inite order spreading model

Extending the classical notion of the spreading model, the $k$-spreading models of a Banach space are introduced, for every $k\in{\mathbb{N}}$. The definition, which is based on the $k$-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of $c_0$ and the composition property are established. As consequence, a problem concerning the structure of the $k$-iterated spreading models is solved.

[^1] [^2] [^3]

Introduction {#introduction .unnumbered}
============

The present work was motivated by a problem of E. Odell and Th. Schlumprecht concerning the structure of the $k$-iterated spreading models of the Banach spaces. Our attempt to answer the problem led to the $k$-spreading models which in turn are based on the $k$-sequences and plegma families. The aim of this paper is to introduce the above concepts and to develop a theory yielding , among others, a solution to the aforementioned problem.

Spreading models, invented by A. Brunel and L. Sucheston (c.f. @BS), posses a key role in the modern Banach space theory. Let us recall that a spreading model of a Banach space $X$ is a spreading sequence[^4] generated by a sequence of $X$. The spreading sequences have regular structure and the spreading models act as the tool for realizing that structure in the space $X$ in an asymptotic manner. This together with the Brunel-Sucheston's discovery that every bounded sequence has a subsequence generating a spreading model determine the significance and importance of this concept. For a comprehensive presentation of the theory of the spreading models we refer the interested reader to the monograph of B. Beauzamy and J.-T. Lapresté (c.f. @BL).

Iteration is naturally applicable to spreading models. Thus one could define the 2-iterated spreading models of a Banach space $X$ to be the spreading sequences which occur as spreading models of the spaces generated by spreading models of $X$. Further iteration yields the $k$-iterated spreading models of $X$, for every $k\in{\mathbb{N}}$. Iterated spreading models appeared in the literature shortly after Brunel-Sucheston's invention. Indeed, B. Beauzamy and B. Maurey in @BM, answering a problem of H.P. Rosenthal, showed that the class of the 2-iterated spreading models does not coincide with the corresponding one of the spreading models. In particular they constructed a Banach space admitting the usual basis of $\ell^1$ as a $2$-iterated spreading model and not as a spreading model.

E. Odell and Th. Schlumprecht in @O-S asked whether or not every Banach space admits a $k$-iterated spreading model equivalent to the usual basis of $\ell^p$, for some $1\leq p<\infty$, or $c_0$. Let us also point out that in the same paper they provided a reflexive space $\mathfrak{X}$ with an unconditional basis such that no $\ell^p$ or $c_0$ is embedded into the space generated by any spreading model of the space. This remarkable result answered a long standing problem of the Banach space theory.

Our approach uses the $k$-spreading models which in many cases include the $k$-iterated ones. The $k$-spreading models are always spreading sequences $(e_n)_n$ in a seminormed space $E$. They are generated by $k$-sequences $(x_s)_{s\in[{\mathbb{N}}]^k}$, where $[{\mathbb{N}}]^k$ denotes the family of all $k$-subsets of ${\mathbb{N}}$. A critical ingredient in the definition is the plegma families $(s_i)_{i=1}^l$ of elements of $[{\mathbb{N}}]^k$, described as follows.

A finite sequence $(s_j)_{j=1}^l$ in $[{\mathbb{N}}]^k$ is a plegma family if its elements satisfy the following order relation: for every $1\leq
i\leq k$, $s_1(i)<\ldots<s_l(i)$ and for every $1\leq i< k$, $s_l(i) < s_1(i+1)$. The plegma families, as they are used in the definition, force a weaker asymptotic relation of the $k$-spreading models to the space $X$, as $k$ increases. For $k=1$, the plegma families coincide to the finite subsets of ${\mathbb{N}}$ yielding that the new definition of the 1-spreading models recovers the classical one. For $k>1$, the plegma families have a quite strict behavior which is described in the first section of the paper. Of independent interest is also Lemma \[intro\_denslem\] stated below.

The $k$-spreading models of a Banach space $X$ are denoted by $\mathcal{SM}_k(X)$ and they define an increasing sequence. As the definition easily yields, the same holds for the $k$-iterated ones. Similarly to the classical case, for every bounded $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ there exists an infinite subset $L$ of ${\mathbb{N}}$ such that the $k$-subsequence $(x_s)_{s\in[L]^k}$ generates a $k$-spreading model.

The advantage of the $k$-spreading models is that, unlike the $k$-iterated ones, for $k\geq2$, the space $X$ determines directly their norm, through the $k$-sequences. Moreover, the $k$-spreading models have a transfinite extension yielding a hierarchy of $\xi$-spreading models for all $\xi<\omega_1$. The definition and the study of this hierarchy is more involved and will be presented elsewhere. We should also mention that L. Halbeisen and E. Odell (c.f. @H-O) introduced the asymptotic models which share some common features with the 2-spreading models. The asymptotic models are associated to bounded 2-sequences $(x_s)_{s\in[{\mathbb{N}}]^2}$ and they are not necessarily spreading sequences.

The paper mainly concerns the definition and the study of the $k$-spreading models. Highlighting the results of the paper we should mention the universal property satisfied by the 2-spreading models of $c_0$. More precisely, it is shown that every spreading sequence is isomorphically equivalent to some 2-spreading model of $c_0$. As the spaces generated by $k$-iterated spreading models of $c_0$ are isomorphic to $c_0$, the previous result shows that the $k$-spreading models do not coincide with the $k$-iterated ones. The composition property is also established. Roughly speaking, under some natural conditions, the $d$-spreading model of a $k$-spreading model of a Banach space $X$ is a $(k+d)$-spreading model of $X$. This result is used for showing that a special class of the $k$-iterated spreading models are actually $k$-spreading models. We also extend to the higher order results of the spreading model theory. Among others we provide conditions for the $k$-sequences to generate unconditional spreading models and we study properties like non-distortion and duality of $\ell^1$ and $c_0$ $k$-spreading models. Moreover we introduce the Cesàro summability for $k$-sequences and we prove the following that extends a classical theorem due to H.P. Rosenthal (c.f. @M [@Ro]).

Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a weakly relatively compact $k$-sequence in $X$, i.e. $\overline{\{x_s:s\in[{\mathbb{N}}]^k\}}^w$ is $w$-compact. Then there exists $M\in [{\mathbb{N}}]^\infty$ such that at least one of the following holds:

1.  The subsequence $(x_s)_{s\in[M]^k}$ generates a $k$-spreading model equivalent to the usual basis of $\ell^1$.

2.  There exists $x_0\in X$ such that for every $L\in
    [M]^\infty$, $(x_s)_{s\in [L]^k}$ is $k$-Cesàro summable to $x_0$.

There are significant differences between the cases $k=1$ and $k\geq2$. First for $k=1$ the two alternatives are exclusive which does not remain valid for $k\geq 2$. Second the proof for the case $k\geq 2$ uses the following density result concerning plegma families which is a consequence of the multidimensional Szemeredi's theorem due to H. Furstenberg and Y. Katznelson (c.f. @FK).

\[intro\_denslem\] Let $\delta>0$ and $k, l\in{\mathbb{N}}$. Then there exists $n_0\in {\mathbb{N}}$ such that for every $n\geq n_0$ and every subset $\mathcal{A}$ of the set of all $k$-subsets of $\{1,\ldots, n\}$ of size at least $\delta (\substack{n\\ k})$, there exists a plegma $l$-tuple $(s_j)_{j=1}^l$ in $\mathcal{A}$.

We close the paper with two examples. The first one is a Banach space similar to the aforementioned one of Odell-Schlumprecht. It is proved that no $k$-spreading model of the space is isomorphic to some $\ell^p$, $1\leq p<\infty$, or $c_0$. The composition property, mentioned above, yields that the same holds for the $k$-iterated spreading models and thus the answer to the aforementioned Odell-Schlumprecht problem is a negative one. In the second example, for every $k\in{\mathbb{N}}$ we present a space $\mathfrak{X}_{k+1}$ admitting the usual basis of $\ell^1$ as a $(k+1)$-spreading model while for every $d\leq k$, $\mathfrak{X}_{k+1}$ does not admit $\ell^1$ as a $d$-spreading model. As we have mentioned, the corresponding problem for $k$-iterated spreading models has been answered in @BM for $k+1=2$. It seems that for $k>1$ this problem is still open. However, recently the $(k+1)$-iterated spreading models have been separated by the $k$ ones in @AM. The proofs in both examples make use of the results exhibited in the previous sections of the paper.

Notation {#notation .unnumbered}
--------

By ${\mathbb{N}}=\{1,2,...\}$ we denote the set of all positive integers. We will use capital letters as $L,M,N,...$ (resp. lower case letters as $s,t,u,...$) to denote infinite subsets (resp. finite subsets) of ${\mathbb{N}}$. For every infinite subset $L$ of ${\mathbb{N}}$, the notation $[L]^\infty$ (resp. $[L]^{<\infty}$) stands for the set of all infinite (resp. finite) subsets of $L$. For every $s\in[{\mathbb{N}}]^{<\infty}$, by $|s|$ we denote the cardinality of $s$. For $L\in[{\mathbb{N}}]^\infty$ and $k\in{\mathbb{N}}$, $[L]^k$ (resp. $[L]^{\leq k}$) is the set of all $s\in[L]^{<\infty}$ with $|s|=k$ (resp. $|s|\leq k$). For every $s,t\in[{\mathbb{N}}]^{<\infty}$, we write $s<t$ if either at least one of them is the empty set, or $\max s<\min t$.

Throughout the paper we shall identify strictly increasing sequences in ${\mathbb{N}}$ with their corresponding range, i.e. we view every strictly increasing sequence in ${\mathbb{N}}$ as a subset of ${\mathbb{N}}$ and conversely every subset of ${\mathbb{N}}$ as the sequence resulting from the increasing ordering of its elements. Thus, for an infinite subset $L=\{l_1<l_2<...\}$ of ${\mathbb{N}}$ and $i\in{\mathbb{N}}$, we set $L(i)=l_i$ and similarly, for a finite subset $s=\{n_1<..<n_k\}$ of ${\mathbb{N}}$ and for $1\leq i\leq k$, we set $s(i)=n_i$. Also, for every $L, N\in
[{\mathbb{N}}]^\infty$ and $s\in [{\mathbb{N}}]^{<\infty}$, we set $L(N)=\{L(N(i)):i\in{\mathbb{N}}\}$ and $L(s)=\{L(s(i)): 1\leq i\leq |s|\}$. Similarly, for every $s\in [{\mathbb{N}}]^{k}$ and $ F\subseteq \{1,...,k\}$, we set $s(F)=\{s(i):i\in F\}$. Also for $1\leq m\leq k$, we set $s|m=\{s(i): 1\leq i\leq m\}$.

For every $s,t\in[{\mathbb{N}}]^{<\infty}$, we write $s\sqsubseteq t$ (resp. $s\sqsubset t$) to denote that $s$ is an initial (resp. *proper* initial) segment of $t$. Given two sequences $(s^1_j)_{j=1}^{l_1}$ and $(s^2_j)_{j=1}^{l_2}$ in $[{\mathbb{N}}]^{<\infty}$, by $(s^1_j)_{j=1}^{l_1\;\;\smallfrown}(s^2_j)_{j=1}^{l_2}$, we denote their concatenation. Similarly for more than two sequences.

For a Banach space $X$ with a Schauder basis $(e_n)_n$ and every $x\in X$, $x=\sum_n \lambda_n
e_n$ we write $\text{supp}(x)$ to denote the support of $x$, i.e. $\text{supp}(x)=\{n\in{\mathbb{N}}:\lambda_n\neq 0\}$. If the support of $x$ is finite and $E\subseteq {\mathbb{N}}$ then by $E(x)$, we denote the restriction of $x$ to $E$, namely $E(x)=\sum_{n\in
E}\lambda_n e_n$.

Two sequences $(x_n)_n$ and $(y_n)_n$, not necessarily in the same Banach space, will be called isometric (resp. equivalent) if (resp. there exists $0<c\leq C$ such that) for every $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ we have that $\|\sum_{i=1}^na_ix_i\|=\|\sum_{i=1}^na_iy_i\|$ (resp. $c\|\sum_{i=1}^na_ix_i\|\leq\|\sum_{i=1}^na_iy_i\|\leq C\|\sum_{i=1}^na_ix_i\|$). Generally concerning Banach space theory the notation and the terminology that we follow is the standard one (see @AK and @Lid-Tza).

Plegma families in $[{\mathbb{N}}]^k$
=====================================

As we have already mentioned, the basic ingredients of the definition of the $k$-spreading models are the $k$-sequences and the plegma families. In this section we introduce the plegma families as well as the related notions of the plegma paths and the plegma preserving maps.

Definition and basic properties {#section
admissibility}
-------------------------------

We start with the definition of the plegma families.

\[defn plegma\] Let $k\in{\mathbb{N}}$ and $M\in [{\mathbb{N}}]^\infty$. A plegma family in $[M]^k$ is a finite sequence $(s_j)_{j=1}^l$ in $[M]^k$ satisfying the following properties.

1.  For every $1\leq
    i\leq k$, $s_1(i)<\ldots<s_l(i)$.

2.  For every $1\leq i< k$, $s_l(i) < s_1(i+1)$.

For each $l\in {\mathbb{N}}$, the set of all sequences $(s_j)_{j=1}^l$ which are plegma families in $[M]^k$ will be denoted by $\textit{Plm}_l([M]^k)$. We also set $\textit{Plm}([M]^k)=\bigcup_{l=1}^\infty\textit{Plm}_l([M]^k)$.

Notice that for $l=1$ and every $k\in {\mathbb{N}}$, we have $\textit{Plm}_1([M]^k)=[M]^k$. Moreover, for $k=1$ and every $l\in{\mathbb{N}}$, $\textit{Plm}_l([M]^1)=[M]^{l}$. In the sequel the elements of $\textit{Plm}_2([M]^k)$ will be called plegma pairs in $[M]^k$.

Although the notion of the plegma family is natural, it does not seem to have appeared in the literature. As it was pointed out to us by S. Todorcevic, a concept that slightly reminds plegma pairs in $[{\mathbb{N}}]^3$ is given by E. Specker in @Sp.

In the next proposition we gather some useful properties of plegma families. The proof is straightforward.

\[rem34\] Let $k,l\in{\mathbb{N}}$, $M\in[{\mathbb{N}}]^\infty$ and $(s_j)_{j=1}^l$ be a finite sequence in $[M]^k$.

1.  $(s_j)_{j=1}^l\in \textit{Plm}_l([M]^k)$ if and only if there exists $F\in[M]^{kl}$ such that $s_j(i)=F((i-1)k+j)$, for every $1\leq i\leq k$ and $1\leq j\leq l$.

2.  If $(s_j)_{j=1}^l\in \textit{Plm}_l([M]^k)$ then $(s_{j_p})_{p=1}^m\in\textit{Plm}_m([M]^k)$, for every $1\leq m\leq l$ and $1\leq j_1<\ldots<j_m\leq l$.

3.  $(s_j)_{j=1}^l\in\textit{Plm}_l([M]^k)$ if and only if $(s_{j_1},s_{j_2})$ is a plegma pair in $[M]^k$, for every $1\leq j_1<j_2\leq l$.

4.  If $(s_j)_{j=1}^l\in\textit{Plm}_l([M]^k)$ then $(s_j(F))_{j=1}^l\in\textit{Plm}_l([M]^{|F|})$, for every non empty $ F\subseteq \{1,...,k\}$.

\[ramseyforplegma\] Let $M$ be an infinite subset of ${\mathbb{N}}$ and $k,l\in{\mathbb{N}}$. Then for every finite partition $\textit{Plm}_l([M]^k)=\bigcup_{j=1}^p
P_j$, there exist $L\in[M]^\infty$ and $1\leq j_0\leq p$ such that $\textit{Plm}_l([L]^k)\subseteq P_{j_0}$.

By Proposition \[rem34\] (i), we conclude that the map sending each plegma family $(s_j)_{j=1}^l$ in $[M]^k$ to its union $\bigcup_{j=1}^ls_j$ is a bijection from $\textit{Plm}_l([M]^k)$ onto $[M]^{kl}$. Therefore the partition of $\textit{Plm}_l([M]^k)$ induces a corresponding one to $[M]^{kl}$ and the conclusion easily follows by applying the Ramsey's theorem @R.

Plegma paths in $[{\mathbb{N}}]^k$
----------------------------------

In this subsection we introduce the definition of the plegma paths. As we shall see in the sequel, the plegma paths play important role in the development of the theory of $k$-spreading models.

Let $l,k\in{\mathbb{N}}$ and $M\in[{\mathbb{N}}]^\infty$. We will say that a finite sequence $(s_j)_{j=0}^l$ is a plegma path of length $l$ from $s_0$ to $s_l$ *in* $[M]^k$, if $(s_{j-1},s_{j})$ is a plegma pair in $[M]^k$, for every $1\leq j\leq l-1$.

\[lemma conserning the length of the plegma path\] Let $k\in{\mathbb{N}}$ and $(s_j)_{j=0}^l$ be a plegma path in $[{\mathbb{N}}]^k$. If $s_0<s_l$ then $l\geq k$.

Suppose on the contrary that $s_0<s_l$ and $l<k$. Since $(s_{j-1}, s_j)$ is a plegma pair in $[{\mathbb{N}}]^k$, we have $s_j(i_1)<s_{j-1}(i_2)$, for every $1\leq j\leq
l$ and $1\leq i_1<i_2\leq k$. Hence, $s_l(1)<s_{l-1}(2)<s_{l-2}(3)<\ldots<s_0(l+1)\leq s_0(k)$, which contradicts that $s_0<s_l$.

Let $k\in{\mathbb{N}}$ and $M\in[{\mathbb{N}}]^\infty$. An $s\in [M]^k$ will be called skipped in $M$ if for every $1\leq i<k$ there exists $m\in
M$ such that $s(i)<m<s(i+1)$. The set of all skipped $s\in [M]^k$ in $M$ will be denoted by $[M]^k_\shortparallel$.

\[rem453\] Notice that for every $m\in{\mathbb{N}}$ and $s\in [M]^k_\shortparallel$ there exists a plegma path $(s_j)_{j=0}^l$ in $[M]^k$ with $s_0=s$.

\[accessing everything with plegma path of length |s\_0|\] Let $k\in{\mathbb{N}}$ and $M\in[{\mathbb{N}}]^\infty$. Then for every $s,t\in[M]_\shortparallel^k$ with $s<t$ there exists a plegma path of length $k$ in $ [M]^k $ from $s$ to $t$. Moreover, every plegma path in $[{\mathbb{N}}]^k$ from $s$ to $t$ has length at least $k$.

Fix $s,t\in[M]_\shortparallel^k$ with $s<t$. It is clear that we may choose $\tilde{s}, \tilde{t}\in [M]^{2k-1}$ such that $\tilde{s}(2i-1)=s(i)$ and similarly $\tilde{t}(2i-1)=t(i)$, for every $1\leq i\leq k$. For every $0\leq j\leq k$, we set $$s_j=\big\{\tilde{s}(2i-1+j):1\leq i\leq k-j\big\}\cup\big\{\tilde{t}(2i-1+k-j):1\leq i\leq j\big\}$$ It is easy to check that $s_0=s$, $s_k=t$ and $(s_j)_{j=0}^k$ is a plegma path in $[M]^k$. Moreover, by Lemma \[lemma conserning the length of the plegma path\], every plegma path in $[{\mathbb{N}}]^k$ from $s$ to $t$ is of length at least $k$. Hence $(s_j)_{j=0}^k$ is a plegma path from $s$ to $t$ in $[M]^k$ with the least possible length and the proof is complete.

\[graphs\] In terms of graph theory the above proposition states that in the directed graph with vertices the elements of $ [{\mathbb{N}}]^k$ and edges the plegma pairs $(s,t)$ in $[{\mathbb{N}}]^k$, the distance between two vertices $s$ and $t$ with $s<t$ is equal to $k$.

Plegma families and mappings
----------------------------

Let $k_1,k_2\in{\mathbb{N}}$, $M\in[{\mathbb{N}}]^\infty$ and $\varphi:[M]^{k_1}\to
  [{\mathbb{N}}]^{k_2}$. We will say that the map $\varphi$ is plegma preserving from $[M]^{k_1}$ into $[{\mathbb{N}}]^{k_2}$ if for every plegma family $(s_j)_{j=1}^l$ in $[M]^{k_1}$, $(\varphi(s_j))_{j=1}^l$ is a plegma family in $[{\mathbb{N}}]^{k_2}$.

\[rem3456\] Let $k_1,k_2\in{\mathbb{N}}$. If $k_1<k_2$ then for every $M\in[{\mathbb{N}}]^\infty$ there exists a plegma preserving map from $[M]^{k_2}$ onto $[M]^{k_1}$. For instance, by Proposition \[rem34\], the map $s\to s|k_1$ is plegma preserving from $[M]^{k_2}$ onto $[M]^{k_1}$.

In contrast to the above remark we have the following.

\[non plegma preserving maps\] Let $k_1,k_2\in{\mathbb{N}}$. If $k_1<k_2$ then for every $M\in[{\mathbb{N}}]^\infty$ and $\varphi:[M]^{k_1}\to [{\mathbb{N}}]^{k_2}$ there exists $L\in[M]^\infty$ such that for every plegma pair $(s_1,s_2)$ in $[ L]^{k_1}$ neither $(\phi(s_1),\phi(s_2))$ nor $(\phi(s_2),\phi(s_1))$ is a plegma pair in $[{\mathbb{N}}]^{k_2}$. In particular, there exists no $L\in[M]^\infty$ such that the map $\varphi$ is plegma preserving from $[L]^{k_1}$ into $[{\mathbb{N}}]^{k_2}$.

Let $M\in[{\mathbb{N}}]^\infty$ and $\varphi:[M]^{k_1}\to [{\mathbb{N}}]^{k_2}$. We set $P_1$ (resp. $P_2$) to be the set of all $(s_1,s_2)\in
\textit{Plm}_2([M]^{k_1})$ such that $(\varphi(s_1),\varphi(s_2))$ (resp. $(\varphi(s_2),\varphi(s_1))$) is a plegma pair in $[{\mathbb{N}}]^{k_2}$ and $P_3=\textit{Plm}_2([M]^{k_1})\setminus (P_1\cup P_2)$. By Theorem \[ramseyforplegma\] there exist $i\in\{1,2,3\}$ and $L\in [M]^\infty$ such that $\textit{Plm}_2([L]^{k_1})\subseteq P_i$. It remains to show that $i=3$.

Indeed, assume that $i= 2$. By Remark \[rem453\] we may choose a plegma path $(s_j)_{j=0}^l$ in $[L]^k$ with $min(\varphi(s_0))<l$. For every $0\leq j\leq l$, we set $n_j=\min(\varphi(s_j))$. Since $\textit{Plm}_2([L]^{k_1})\subseteq P_2$, we have that $(n_j)_{j=0}^l$ is a strictly decreasing sequence in ${\mathbb{N}}$ with length $l+1$. Since $n_0<l$ this is impossible.

It remains to show that $i\neq 1$. Indeed, assume on the contrary. Then notice that $\varphi$ transforms every plegma path in $[L]^{k_1}$ to a plegma path of equal length in $[{\mathbb{N}}]^{k_2}$. Using Remark \[rem453\], it is easy to see that we may choose $s<t$ in $[L]_\shortparallel^{k_1}$ such that $\varphi(s)<\varphi(t)$ and $\varphi(s), \varphi(t)\in
[{\mathbb{N}}]_\shortparallel^{k_2}$. By Proposition \[accessing everything with plegma path of length |s\_0|\] and Remark \[graphs\], we have that the distance of $s,t$ is equal to $k$ while that of $\varphi(s), \varphi(t)$ is equal to $k_2$. But since $s,t$ are joined by a plegma path of length $k_1$ and $\varphi$ preserves plegma paths we have that the distance of $\varphi(s), \varphi(t)$ is at most $k_1$. Hence $k_2\leq k_1$, a contradiction.

\[lemma making a hereditary nonconstant function, nonconstant on plegma pairs\] Let $A$ be a set, $k\in{\mathbb{N}}$, $M\in[{\mathbb{N}}]^\infty$ and $\varphi:[M]^k\to A$. Then there exists $L\in [M]^\infty$ such that either the restriction of $\varphi$ on $[L]^k$ is constant or for every plegma pair $(s_1,s_2)$ in $[ L]^k$, $\varphi(s_1)\neq\varphi(s_2)$.

By Theorem \[ramseyforplegma\] there exists $N\in [M]^\infty$ such that exactly one of the following are satisfied.

1.  For every plegma pair $(s_1,s_2)$ in $[N]^k$, $\varphi(s_1)=\varphi(s_2)$.

2.  For every plegma pair $(s_1,s_2)$ in $[ N]^k$, $\varphi(s_1)\neq\varphi(s_2)$.

Therefore, it suffices to show that the first alternative implies that there exists $L\in [N]^\infty$ such that $\varphi$ is constant on $[L]^k$. Indeed, let $s=(N(2),N(4),...,N(2k))$, $L=\{N(2n):
n\geq k+1\}$ and $t\in [L]^k$. Observe that $s<t$ and $s,t\in[N]^k_\shortparallel$ and therefore, by Proposition \[accessing everything with plegma path of length |s\_0|\], there exists a plegma path $(s_j)_{j=0}^k$ of length $k$ in $[N]^k$ with $s_0=s$ and $s_k=t$. Assuming that (i) holds, we get that $$\varphi(s)=\varphi(s_0)=\varphi(s_1)=\ldots=\varphi(s_k)=\varphi(t)$$ Hence for every $t\in[L]^k$, $\varphi(t)=\varphi(s)$, i.e. $\varphi$ is constant on $[L]^k$.

Spreading sequences
===================

We recall that a sequence $(e_n)_{n}$ in a seminormed linear space $(E,\|\cdot\|_*)$ is called *spreading* if it is isometric to any of its subsequences, i.e. for every $n\in{\mathbb{N}}$, $a_1,\ldots,a_n\in {\mathbb{R}}$ and $k_1<\ldots<k_n$ in ${\mathbb{N}}$ we have that $\|\sum_{j=1}^na_je_j\|_*=\|\sum_{j=1}^na_je_{k_j}\|_*$. In this section we will briefly discuss the norm properties of the spreading sequences. The interested reader can find a detailed analysis in the monographs @AK and @BL.

The proof of the following result shares similar ideas with the one of Proposition I.1.B.2 in @BL.

\[sing\] Let $(E,\|\cdot\|_*)$ be a seminormed linear space and $(e_n)_{n}$ be a spreading sequence in $E$. Then the following are equivalent.

1.  There exist $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ not all zero, with $\|\sum_{i=1}^na_ie_i\|_*=0$.

2.  For every $n,m\in{\mathbb{N}}$, $\|e_n-e_m\|_*=0$.

3.  For every $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$, $
    \|\sum_{i=1}^na_ie_i\|_*=|\sum_{i=1}^na_i|\cdot \|e_1\|_*$.

Spreading sequences in seminormed linear spaces satisfying (i)-(iii) of the above proposition will be called *trivial*. By (i) we have that if $(e_n)_{n}$ is non trivial, then $(e_n)_{n}$ is linearly independent and the restriction of the seminorm $\|\cdot\|_*$ to the linear subspace of $E$ generated by $(e_n)_n$ is actually norm. Therefore, every non trivial spreading sequence generates a Banach space.

We classify the non trivial spreading sequences into the following three categories:

1.  The *singular* spreading sequences, i.e. the non trivial spreading sequences which are not Schauder basic sequences.

2.  the *unconditional* spreading sequences and

3.  the *conditional Schauder basic* spreading sequences, i.e. the non trivial spreading sequences which are Schauder basic but not unconditional.

The next two results are restatements of Propositions I.1.4 and I.4.2 of @BL respectively.

\[equiv forms for 1-subsymmetric weakly null\] Let $(e_n)_{n}$ be a non trivial spreading sequence. Then the following are equivalent.

1.  $(e_n)_{n}$ is unconditional and not equivalent to the usual basis of $\ell^1$.

2.  $(e_n)_{n}$ is weakly null.

3.  $(e_n)_{n}$ is Cesáro summable to zero.

4.  $(e_n)_{n}$ is 1-unconditional and not equivalent to the usual basis of $\ell^1$.

\[thmsingular\] Let $(e_n)_n$ be a non trivial spreading sequence and $E$ the Banach space generated by $(e_n)_n$. Then $(e_n)_n$ is singular if and only if $(e_n)_n$ is weakly convergent to a nonzero element $e\in
E$.

\[properties of the natural decomposition\] Let $(e_n)_n$ be a singular spreading sequence. By the above proposition, we have that $(e_n)_n$ is of the form $e_n=e'_n+e$, where $e$ is nonzero and $(e'_n)_n$ is weakly null. This decomposition of $(e_n)_n$ as $e_n=e'_n+e$ will be called *the natural decomposition* of $(e_n)_n$. It is easy to check that $(e'_n)_{n}$ is non trivial, spreading and not equivalent to the usual basis of $\ell^1$. Hence by Proposition \[thmsingular\], $(e'_n)_n$ is unconditional, weakly null and Cesàro summable to zero. Moreover, if $E$ and $E'$ are the Banach spaces generated by the sequences $(e_n)_n$ and $(e'_n)_n$ respectively, then $E,E'$ are isomorphic and $E=E'\oplus<e>$.

Finally for the conditional Schauder basic spreading sequences we have the next characterization, which is a consequence of the above results and Rosenthal's $\ell^1$ theorem @Ro.

Let $(e_n)_n$ be a spreading non trivial sequence and $E$ be the Banach space generated by $(e_n)_n$. Then $(e_n)_n$ is a conditional Schauder basic sequence if and only if $(e_n)_n$ is non trivial weak-Cauchy.

$k$-sequences and $k$-spreading models
======================================

In this section we present the definition of the $k$-sequences and we introduce the notion of the $k$-spreading models, for all $k\in{\mathbb{N}}$. As we will see, for $k=1$, the definition coincides with the classical one of A. Brunel and L. Sucheston @BS.

Definitions and basic properties
--------------------------------

We start with the definition of the $k$-sequences.

Let $k\in{\mathbb{N}}$ and $X$ be a non empty set. A $k$-sequence in $X$ is a map $\varphi:[{\mathbb{N}}]^k\to X$. A $k$-subsequence in $X$ is a map of the form $\varphi:[M]^k\to X$ , where $M\in [{\mathbb{N}}]^\infty$.

A $k$-sequence $\varphi:[{\mathbb{N}}]^k\to X$ will be usually denoted by $(x_s)_{s\in [{\mathbb{N}}]^k}$, where $x_s=\varphi(s)$, $s\in [{\mathbb{N}}]^k$. Similarly, the notation $(x_s)_{s\in [M]^k}$ stands for the $k$-subsequences $\varphi:[M]^k\to X$.

\[Definition of spreading model\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$, $(x_s)_{s\in [{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $(E,\|\cdot\|_*)$ be an infinite dimensional seminormed linear space with Hamel basis $(e_n)_{n}$. Also let $M\in[{\mathbb{N}}]^\infty$ and $(\delta_n)_n$ be a null sequence of positive reals. We will say that the $k$-subsequence $(x_s)_{s\in [M]^k}$ generates $(e_n)_{n}$ as a spreading model as a $k$-spreading model (with respect to $(\delta_n)_n$), if the following is satisfied.

For every $m,l\in {\mathbb{N}}$, with $m\leq l$, every $(s_j)_{j=1}^m\in\textit{Plm}_m([M]^k)$ with $s_1(1)\geq M(l)$ and every choice of $a_1,...,a_m\in[-1,1]$, we have $$\label{rsm}\Bigg{|}\Big{\|}\sum_{j=1}^m a_j
x_{s_j}\Big{\|}-\Big{\|}\sum_{j=1}^m a_j
  e_j\Big{\|}_* \Bigg{|}\leq\delta_l$$

Since $\text{Plm}([{\mathbb{N}}]^1)=[{\mathbb{N}}]^{<\infty}$, it is clear that for $k=1$, Definition \[Definition of spreading model\] coincides with the classical definition of a spreading model of an ordinary sequence $(x_n)_n$ in a Banach space $X$. Thus the $1$-spreading models are the usual ones. Moreover, it is easy to see that for every $k\in{\mathbb{N}}$, every $k$-spreading model $(e_n)_{n}$ is a spreading sequence.

Let's point out here that there exist $k$-sequences in Banach spaces which generate $k$-spreading models which are trivial spreading sequences, in other words (see Proposition \[sing\]), $\|\cdot\|_*$ is not a norm. For instance, this occurs for every constant $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$. We should also point out that even if $(e_n)_n$ is non trivial, it is not necessarily a Schauder basic sequence. More information on this issue are contained in Section \[s5\].

In the next proposition we state some stability properties of the $k$-spreading models. The proof is straightforward.

\[remark on the definition of spreading model\] Let $k\in{\mathbb{N}}$, $(x_s)_{s\in [{\mathbb{N}}]^k}$ be a $k$-sequence in a Banach space $X$, $M\in[{\mathbb{N}}]^\infty$ and $(\delta_n)_n$ be a null sequence of positive reals. If $(x_s)_{s\in [M]^k}$ generates a sequence $(e_n)_{n}$ as a $k$-spreading model with respect to $(\delta_n)_n$ then the following are satisfied.

1.  For every $L\in[M]^\infty$, $(x_s)_{s\in [L]^k}$ generates $(e_n)_{n}$ as a $k$-spreading model with respect to $(\delta_n)_n$.

2.  For every null sequence $(\delta'_n)_{n}$ of positive reals there exists $M'\in[M]^\infty$ such that $(x_s)_{s\in [M']^k}$ generates $(e_n)_{n}$ as a $k$-spreading model with respect to $(\delta'_n)_{n}$.

3.  The $k$-sequence $(y_s)_{s\in[{\mathbb{N}}]^k}$, defined by $y_s=x_{M(s)}$, $s\in[{\mathbb{N}}]^k$, generates $(e_n)_{n}$ as a $k$-spreading model with respect to $(\delta_n)_n$.

Let us also notice that for $k=1$ the assertion that (\[rsm\]) holds for all $m\leq l$ is redundant. This is not the case for $k\geq 2$, since a plegma family in $[{\mathbb{N}}]^k$ is not always a subsequence of a larger one. However, the next lemma shows that we may bypass this extra condition by passing to a sparse infinite subset of ${\mathbb{N}}$.

\[old defn yields new\] Let $k\in{\mathbb{N}}$, $(x_s)_{s\in [{\mathbb{N}}]^k}$ be a $k$-sequence in a Banach space $X$, $L\in[{\mathbb{N}}]^\infty$, $(E,\|\cdot\|_*)$ be an infinite dimensional seminormed linear space with Hamel basis $(e_n)_{n}$ and $(\delta_n)_{n}$ be a null sequence of positive reals such that $$\label{ewq}\Bigg{|}\Big{\|}\sum_{j=1}^l a_j
x_{t_j}\Big{\|}-\Big{\|}\sum_{j=1}^l a_j e_j\Big{\|}_*
\Bigg{|}\leq\delta_l$$ for every $l\in{\mathbb{N}}$, every $(t_j)_{j=1}^l\in\textit{Plm}_l([ L]^k)$ with $t_1(1)\geq L(l)$ and every choice of $a_1,...,a_l\in[-1,1]$. Then there exists $M\in[L]^\infty$ such that $(x_s)_{s\in [M]^k}$ generates $(e_n)_{n}$ as a $k$-spreading model with respect to $(\delta_n)_n$.

We choose $M\in [L]^\infty$ such that for every $l\in{\mathbb{N}}$ there exist at least $l-1$ elements of $L$ between $M(l)$ and $M(l+1)$. Then notice that for every $m,l\in{\mathbb{N}}$ with $m\leq l$ and every $(s_j)_{j=1}^m\in\text{Plm}_m([M]^k)$ with $s_j(1)\geq M(l)$, there exists $(t_j)_{j=1}^l\in\text{Plm}_l([L]^k)$ with $s_j=t_j$ for all $1\leq j\leq m$. This observation and (\[ewq\]) easily yield that for every $m,l\in {\mathbb{N}}$, with $m\leq l$, every $(s_j)_{j=1}^m\in\textit{Plm}_m([M]^k)$ with $s_1(1)\geq M(l)$ and every choice of $a_1,...,a_m\in[-1,1]$, we have $$\Bigg{|}\Big{\|}\sum_{j=1}^m a_j
x_{s_j}\Big{\|}-\Big{\|}\sum_{j=1}^m a_j
 e_j\Big{\|}_* \Bigg{|}\leq\delta_l$$ and the proof is complete.

Existence of $k$-speading models
--------------------------------

In this subsection we will show that every bounded $k$-sequence in a Banach space $X$ contains a $k$-subsequence which generates a $k$-spreading model. The proof follows similar lines with the corresponding one of the classical spreading models.

For $k\in{\mathbb{N}}$ and a $k$-sequence $(x_s)_{s\in
[{\mathbb{N}}]^k}$ in a Banach space $X$, we will say that $(x_s)_{s\in
[{\mathbb{N}}]^k}$ *admits* $(e_n)_{n}$ as a $k$-spreading model (or $(e_n)_{n}$ *is* a $k$-spreading model of $(x_s)_{s\in
[{\mathbb{N}}]^k}$) if there exists $M\in[{\mathbb{N}}]^\infty$ such that the subsequence $(x_s)_{s\in [M]^k}$ generates $(e_n)_{n}$ as a $k$-spreading model. A $k$-sequence $(x_s)_{s\in
[{\mathbb{N}}]^k}$ in $X$ will be called *bounded* (resp. *seminormalized*) if there exists $C>0$ (resp. $0<c\leq C$) such that $\|x_s\|\leq C$ (resp. $c\leq\|x_s\|\leq C$), for every $s\in [{\mathbb{N}}]^k$.

For all $k\in {\mathbb{N}}$, every bounded $k$-sequence in a Banach space $X$ admits a $k$-spreading model.

Let $X$ be a Banach space and $k\in{\mathbb{N}}$, $(x_s)_{s\in [{\mathbb{N}}]^k}$ be a bounded $k$-sequence in $X$. We divide the proof into four steps.

**Step 1.** Let $l\in{\mathbb{N}}$, $N\in[{\mathbb{N}}]^\infty$ and $\delta>0$. Then there exists $L\in[N]^\infty$ such that $$\Bigg{|}\Big{\|}\sum_{j=1}^la_jx_{t_j}\Big{\|}-\Big{\|}\sum_{j=1}^la_jx_{s_j}\Big{\|}\Bigg{|}\leq\delta$$ for every $(t_j)_{j=1}^l, (s_j)_{j=1}^l
\in\textit{Plm}_l([L]^k)$ and $a_1,...,a_l\in[-1,1]$.

\[Proof of Step 1:\] Let $(\textbf{a}_i)_{i=1}^{n_0}$ be a $\frac{\delta}{3l}-$net of the unit ball of $\big({\mathbb{R}}^l, \|\cdot\|_\infty\big)$. We set $N_0=N$. By a finite induction on $1\leq i\leq n_0$, we construct a decreasing sequence $N_0\supseteq N_1\supseteq\ldots\supseteq N_{n_0}$ as follows. Suppose that $N_0,\ldots,N_{i-1}$ have been constructed. Let $\textbf{a}_i=(a_j^i)_{j=1}^l$ and $g_i:\textit{Plm}_l([N_{i-1}]^k)\to[0,lC]$ defined by $g_i\big((s_j)_{j=1}^l\big)=\|\sum_{j=1}^l a_j^i x_{s_j}\|$. We partition the interval $[0,lC]$ into disjoint intervals of length $\frac{\delta}{3}$ and applying Theorem \[ramseyforplegma\] we find $N_i\in[N_{i-1}]^\infty$ such that for every $(t_j)_{j=1}^l,(s_j)_{j=1}^l\in\textit{Plm}_l( [N_i]^k)$, we have $|g_i((t_j)_{j=1}^l)-g_i((s_j)_{j=1}^l)|<\frac{\delta}{3}$. Proceeding in this way we conclude that for every $(s_j)_{j=1}^l,(t_j)_{j=1}^l\in\textit{Plm}_l([ N_{n_0}]^k)$ and $1\leq i\leq n_0$, we have that $\Big{|}\|\sum_{j=1}^la_j^ix_{t_j}\|-\|\sum_{j=1}^la_j^ix_{s_j}\|\Big{|}\leq\frac{\delta}{3}$. Since $(\textbf{a}_i)_{i=1}^{n_0}$ is a $\frac{\delta}{3}-$net of the unit ball of $({\mathbb{R}}^l, \|\cdot\|_\infty)$ it is easy to see that $L=N_{n_0}$ is as desired.

**Step 2.** Let $(\delta_n)_n$ be a null sequence of positive real numbers. Then there exists $M\in[{\mathbb{N}}]^\infty$ such that for every $m\leq l$, every $(t_j)_{j=1}^m, (s_j)_{j=1}^m
\in\textit{Plm}_m([M]^k)$ with $s_1(1),t_1(1)\geq M(l)$ and $a_1,...,a_m\in[-1,1]$, we have $$\label{equo}\Bigg{|}\Big\|\sum_{j=1}^m a_jx_{t_j}\Big\|-\Big\|\sum_{j=1}^m
a_jx_{s_j}\Big\|\Bigg{|}\leq\delta_l$$

\[Proof of Step 2:\] By Step 1 and a standard diagonalization we easily obtain an $L\in [{\mathbb{N}}]^\infty$ satisfying $\Big{|}\|\sum_{j=1}^l a_jx_{t_j}\|-\|\sum_{j=1}^l
a_jx_{s_j}\|\Big{|}\leq\delta_l$, for every $ l\in{\mathbb{N}}$, every $(t_j)_{j=1}^l, (s_j)_{j=1}^l \in\textit{Plm}_l([L]^k)$ with $s_1(1),t_1(1)\geq L(l)$ and $a_1,...,a_l\in[-1,1]$. By Lemma \[old defn yields new\], there exists $M\in [L]^\infty$ satisfying (\[equo\]).

**Step 3.** Let $M\in[{\mathbb{N}}]^\infty$ be the resulting from Step 2 infinite subset of ${\mathbb{N}}$. Also let $l\in{\mathbb{N}}$ and $a_1,...,a_l\in{\mathbb{R}}$. Then for every sequence $\big((s_j^n)_{j=1}^l\big)_{n}$ with $(s_j^n)_{j=1}^l\in\text{Plm}_l([M]^k)$, for all $n\in{\mathbb{N}}$ and $\lim s^n_1(1)=+\infty$, the sequence $(\|\sum_{j=1}^la_jx_{s_j}^n\|)_{n}$ is a Cauchy sequence in $[0,+\infty)$. Moreover, $\lim_n\|\sum_{j=1}^la_jx_{s_j}^n\|$ is independent from the choice of the sequence $((s_j^n)_{j=1}^l)_{n}$.

\[Proof of Step 3:\] It is straightforward by Step 2.

**Step 4.** Let $(e_n)_n$ be the natural Hamel basis of $c_{00}({\mathbb{N}})$. For every $l\in{\mathbb{N}}$ and $a_1,...,a_l\in{\mathbb{R}}$, we define $$\Big\|\sum_{j=1}^l
a_je_j\Big\|_*=\lim_n\|\sum_{j=1}^la_jx_{s_j}^n\|$$ where for every $n\in{\mathbb{N}}$, $(s_j^n)_{j=1}^l\in\text{Plm}_l([M]^k)$ and $\lim
s^n_1(1)=+\infty$. Then $\|\cdot\|_*$ is a seminorm on $c_{00}({\mathbb{N}})$ under which the natural Hamel basis $(e_n)_n$ is a spreading sequence. Moreover for all $ m\leq l$, $a_1,\ldots, a_m\in[-1,1]$ and $(s_j)_{j=1}^m\in\textit{Plm}_m([M]^k)$ with $s_1(1)\geq
M(l)$, we have $\Big{|}\|\sum_{j=1}^m a_jx_{s_j}\|-\|\sum_{j=1}^m
a_je_j\|_*\Big{|}\leq\delta_l$.

\[Proof of Step 4:\] It follows easily by Steps 2 and 3.

By Step 4, we have that $(x_s)_{s\in [M]^{k}}$ generates $(e_n)_n$ as a $k$-spreading model and the proof is complete.

The increasing hierarchy of $k$-spreading models
------------------------------------------------

In this subsection we will show that the $k$-spreading models of a Banach space $X$ form an increasing hierarchy.

We start with the following lemma which is an easy consequence of Remark \[rem3456\].

\[propxi\] Let $k_1,k_2\in{\mathbb{N}}$ with $1\leq k_1<k_2$. Let $X$ be a Banach space and $(w_t)_{t\in [{\mathbb{N}}]^{k_1}}$ be a $k_1$-sequence in $X$. Let $(x_s)_{s\in [{\mathbb{N}}]^{k_2}}$ be the $k_2$-sequence in $X$ defined by $x_s=w_{s|k_1}$, for every $s\in  [{\mathbb{N}}]^{k_2}$. Then $(w_t)_{t\in
[{\mathbb{N}}]^{k_1}}$ and $(x_s)_{s\in [{\mathbb{N}}]^{k_2}}$ admit the same $k$-spreading models.

For a subset $A$ of $X$ we will say that $A$ *admits* $(e_n)_n$ *as a* $k$-spreading model (or $(e_n)_{n}$ *is* a $k$-spreading model of $A$) if there exists a $k$-sequence $(x_s)_{s\in [{\mathbb{N}}]^k}$ in $A$ which admits $(e_n)_{n}$ as a $k$-spreading model.

Let $X$ be a Banach space, $A\subseteq X$ and $k\in{\mathbb{N}}$. The set of all $k$-spreading models of $A$ will be denoted by $\mathcal{SM}_k(A)$.

By Lemma \[propxi\], we easily obtain the following.

\[incresspr\] Let $X$ be a Banach space and $A\subseteq X$. Then for all $k_1,k_2\in{\mathbb{N}}$ with $k_1< k_2$, we have $\mathcal{SM}_{k_1}(A)\subseteq \mathcal{SM}_{k_2}(A)$,

In Section \[s12\], for each $k\in{\mathbb{N}}$, we construct a Banach space $\mathfrak{X}_{k+1}$ such that $\mathcal{SM}_{k}(\mathfrak{X}_{k+1})\subsetneqq\mathcal{SM}_{k+1}(\mathfrak{X}_{k+1})$. Here, we present a much simpler example of a space $X$ and a proper subset $A$ of $X$ satisfying $\mathcal{SM}_{k}(A)\subsetneqq\mathcal{SM}_{k+1}(A)$.

\[example\] Let $(e_n)_{n}$ be a normalized spreading and $1$-unconditional sequence in a Banach space $(E,\|\cdot\|)$ which is not equivalent to the usual basis of $c_0$. Let $k\in{\mathbb{N}}$ and $(x_s)_{s\in [{\mathbb{N}}]^{k+1}}$ be the natural Hamel basis of $c_{00}([{\mathbb{N}}]^{k+1})$. For $x\in c_{00}([{\mathbb{N}}]^{k+1})$ we define $$\|x\|_{k+1}=\sup\Big{\{}\Big\|\sum_{i=1}^lx(s_i)e_i\Big\|:
l\in{\mathbb{N}},(s_i)_{i=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^{k+1})\text{ and
}s_1(1)\geq l\Big{\}}$$ We set $X=\overline{(c_{00}([{\mathbb{N}}]^{k+1}),\|\cdot\|_{k+1})}$ and $A=\{x_s:s\in [{\mathbb{N}}]^{k+1}\}$. It is easy to see that the sequence $(e_n)_{n}$ is generated by $(x_s)_{s\in[{\mathbb{N}}]^{k+1}}$ as a $(k+1)$-spreading model and thus it belongs to $\mathcal{SM}_{k+1}(A)$. We shall show that for every $(\tilde{e}_n)_{n}\in\mathcal{SM}_k(A)$, either $(\tilde{e}_n)_{n}$ is a trivial spreading sequence or it is isometric to the usual basis of $c_0$. Therefore, there is no sequence in $\mathcal{SM}_{k}(A)$ equivalent to $(e_n)_n$.

Indeed, let $(\tilde{e}_n)_{n}\in\mathcal{SM}_k(A)$. By Proposition \[remark on the definition of spreading model\], we may assume that there exists a $k$-sequence in $A$, $(y_t)_{t\in [{\mathbb{N}}]^k}$ which generates $(\tilde{e}_n)_{n}$ as a $k$-spreading model. Let $\varphi:[{\mathbb{N}}]^k\to [{\mathbb{N}}]^{k+1}$ such that $y_t=x_{\varphi(t)}$, for all $t\in [{\mathbb{N}}]^{k}$. By Proposition \[lemma making a hereditary nonconstant function, nonconstant on plegma pairs\], there exists $M\in [{\mathbb{N}}]^\infty$ such that either $\varphi$ is constant on $[M]^{k}$ or for every plegma pair $(t_1,t_2)$ in $[M]^{k}$, $\varphi(t_1)\neq\varphi(t_2)$. By Proposition \[remark on the definition of spreading model\], we have that $(y_t)_{t\in [M]^k}$ also generates $(\tilde{e}_n)_{n}$ as a $k$-spreading model.

If $\varphi$ is constant on $[M]^k$ then $(\tilde{e}_n)_{n}$ is a trivial sequence. Otherwise, by Theorem \[non plegma preserving maps\], there exists $L\in[M]^\infty$ such that for every plegma pair $(t_1,t_2)$ in $[L]^k$ neither $(\varphi(t_1),\varphi(t_2))$, nor $(\varphi(t_2),\varphi(t_1))$ is a plegma pair in $[{\mathbb{N}}]^{k+1}$. Therefore, for every $(t_j)_{j=1}^m\in\textit{Plm}([L]^k)$ and $(s_j)_{j=1}^l\in\textit{Plm}([{\mathbb{N}}]^{k+1})$ there is at most one $j\in\{1,...,m\}$ and at most one $i\in\{1,...,l\}$ with $\varphi(t_j)=s_i$. This observation and the definition of the norm $\|\cdot\|_{k+1}$, easily implies that $$\label{eqwert}\Big\|\sum_{j=1}^ma_jy_{t_j}\Big\|_{k+1}
=\Big\|\sum_{j=1}^ma_jx_{\varphi(t_j)}\Big\|_{k+1}=\max_{1\leq
j\leq m}|a_j|$$ for all $m\in{\mathbb{N}}$, $a_1,\ldots,a_m\in{\mathbb{R}}$ and $(t_j)_{j=1}^m\in\textit{Plm}([L]^k)$. Since $L\in [M]^\infty$, we have that $(\tilde{e}_n)_{n}$ is generated by $(y_t)_{t\in[L]^k}$ and by (\[eqwert\]), the sequence $(\tilde{e}_n)_{n}$ is isometric to the usual basis of $c_0$.

Topological properties of $k$-sequences
=======================================

This section is devoted to the study of the $k$-sequences in a topological space. We define the convergence of the $k$-sequences in a topological space and we introduce the notion of the subordinated $k$-sequences.

Convergence of $k$-sequences in topological spaces
--------------------------------------------------

We start with the following natural extension of the notion of convergence of sequences in topological spaces.

\[defn convergence of f-sequences\] Let $(X,{\mathcal{T}})$ be a topological space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$. Also let $M\in[{\mathbb{N}}]^\infty$ and $x_0\in X$. We will say that $(x_s)_{s\in[M]^k}$ converges to $x_0$ if for every $U\in{\mathcal{T}}$ with $x_0\in U$ there exists $m\in {\mathbb{N}}$ such that for every $s\in
[M]^k$ with $ s(1)\geq M(m)$ we have that $x_s\in U$.

It is straightforward that if a $k$-subsequence $(x_s)_{s\in[M]^k}$ in a topological space is convergent to some $x_0\in X$, then every further $k$-subsequence of $(x_s)_{s\in
[M]^k}$ is also convergent to $x_0$. Moreover, every continuous map between two topological spaces preserves the convergence of $k$-sequences, i.e. if $\phi:(X_1,\mathcal{T}_1)\to
(X_2,\mathcal{T}_2)$ is continuous and $(x_s)_{s\in[M]^k}$ converges to $x_0\in X_1$, then $(\phi(x_s))_{s\in[M]^k}$ converges to $\phi(x_0)\in X_2$.

However, for $k\geq 2$, there are some differences with the ordinary convergent sequences in topological spaces. For instance it is easy to see that for $k\geq 2$, the convergence of a $k$-sequence $(x_s)_{s\in[M]^k}$ to some $x_0\in X$, does not in general imply that the set $\{x_s:s\in[M]^k\}$ is relatively compact.

Subordinated $k$-sequences
--------------------------

In this subsection we introduce the definition of the subordinated $k$-sequences in a topological space. First, recall that the powerset of ${\mathbb{N}}$ is naturally identified with $\{0,1\}^{\mathbb{N}}$. In this way, for all $k\in{\mathbb{N}}$ and $M\in [{\mathbb{N}}]^\infty$, the set $[M]^{\leq k}$ becomes a compact metric space containing $[M]^k$ as a dense subspace. Moreover, notice that an element $s\in [M]^{\leq k}$ is isolated in $[M]^{\leq k}$ if and only if $s\in [M]^k$.

\[defn subordinating\] Let $(X,{\mathcal{T}})$ be a topological space, $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $M\in[{\mathbb{N}}]^\infty$. We say that $(x_s)_{s\in[M]^k}$ is subordinated $($with respect to $(X,\mathcal{T})$$)$ if there exists a continuous map $\widehat{\varphi}:[M]^{\leq k}\to (X,\mathcal{T})$ such that $\widehat{\varphi}(s)=x_s$, for all $s\in[M]^k$.

\[remark on ff subordinated\] If $(x_s)_{s\in[M]^k}$ is subordinated, then there exists a unique continuous map $\widehat{\varphi}:[M]^{\leq k}\to (X,\mathcal{T})$ witnessing this. Indeed, this is a consequence of the fact that $[M]^k$ is dense in $[M]^{\leq k}$. Also, $\overline{\{x_s:s\in[M]^k\}}=\widehat{\varphi}\big([M]^{\leq
k}\big),$ where $\overline{\{x_s:s\in[M]^k\}}$ is the closure of $\{x_s : s\in[M]^k\}$ in $X$ with respect to $\mathcal{T}$. Therefore, $\overline{\{x_s:s\in[M]^k\}}$ is a countable compact metrizable subspace of $(X,{\mathcal{T}})$ with Cantor-Bendixson index at most $k+1$. Also notice that if $(x_s)_{s\in[M]^k}$ is subordinated then $(x_s)_{s\in[L]^k}$ is also subordinated, for every $L\in
[M]^\infty$.

\[subordinating yields convergence\] Let $(X,{\mathcal{T}})$ be a topological space, $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $M\in[{\mathbb{N}}]^\infty$. Suppose that $(x_s)_{s\in[M]^k}$ is subordinated and let $\widehat{\varphi}:[M]^{\leq k}\to
(X,\mathcal{T})$ be the continuous map witnessing this. Then $(x_s)_{s\in[M]^k}$ is convergent to $\widehat{\varphi}(\emptyset)$.

Let $(y_s)_{s\in [M]^k}$ be the $k$-sequence in $[M]^k$, with $y_s=s$, for all $s\in [M]^k$. Notice that $(y_s)_{s\in [M]^k}$ converges to the empty set and since $\widehat{\varphi}:[M]^{\leq k}\to
(X,\mathcal{T})$ is continuous, we have that $\big(\widehat{\varphi}(y_s)\big)_{s\in [M]^k}$ converges to $\widehat{\varphi}(\emptyset)$. Since $\widehat{\varphi}(y_s)=\widehat{\varphi}(s)=x_s$, for all $s\in
[M]^k$, we conclude that $(x_s)_{s\in[M]^k}$ is convergent to $\widehat{\varphi}(\emptyset)$.

\[Create subordinated\] Let $(X,\mathcal{T})$ be a topological space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$. Then for every $N\in[{\mathbb{N}}]^\infty$ such that $\overline{\{x_s:\;s\in[N]^k\}}$ is a compact metrizable subspace of $(X,{\mathcal{T}})$ there exists $M\in[N]^\infty$ such that $(x_s)_{s\in[M]^k}$ is subordinated.

The proposition obviously holds for $k=1$, since in this case, subordinated and convergent sequences coincide. We proceed by induction on $k\in{\mathbb{N}}$. Assume that Proposition \[Create subordinated\] holds for some $k\in{\mathbb{N}}$ and let $(x_s)_{s\in
[N]^{k+1}}$ be a $(k+1)$-sequence in $X$. Let $N\in [{\mathbb{N}}]^\infty$ such that $\overline{\{x_s:\;s\in[N]^{k+1}\}}$ is a compact metrizable subspace of $(X,{\mathcal{T}})$. We also fix a compatible metric $d$ of $\overline{\{x_s:\;s\in[N]^{k+1}\}}$.

Inductively we choose a strictly increasing sequence $(l_n)_n$ in ${\mathbb{N}}$, a decreasing sequence $(L_n)_{n}$ of infinite subsets of $N$ and a $k$-sequence $(x_s)_{s\in[L]^k}$ in $X$, where $L=\{l_n:
n\in{\mathbb{N}}\}$ such that for every $n\in{\mathbb{N}}$, the following are satisfied.

1.  $l_n<\min L_n$.

2.  For every $l\in L_n$ and every $t\in[\{l_1,...,l_n\}]^k $, $(x_{t\cup\{l\}})_{l\in
    L_n}\to x_t$ and in addition if $\max t=l_n$, then $d(x_{t\cup\{l\}},x_t)<\frac{1}{n}$.

We omit the construction since it is straightforward. By the inductive assumption there exists $M\in[L]^\infty$ such that $(x_t)_{t\in[M]^k}$ is subordinated. If $\widehat{\psi}:[M]^{\leq
k}\to X$ is the continuous map witnessing this then we extend $\widehat{\psi}$ to the map $\widehat{\varphi}:[M]^{\leq k+1}\to X$, by setting $\widehat{\varphi}(s)=x_s$, for every $s\in[M]^{k+1}$. Using condition (ii), we easily show that $\widehat{\varphi}$ is continuous and therefore $(x_s)_{s\in[M]^{k+1}}$ is subordinated.

By Propositions \[subordinating yields convergence\] and \[Create subordinated\], we have that every $k$-sequence in a compact metrizable space contains a convergent $k$-subsequence.

Weakly relatively compact $k$-sequences in Banach spaces
========================================================

It is well known that for every sequence $(x_n)_n$ in a weakly compact subset of a Banach space $X$ there exists $M\in {\mathbb{N}}$ such that the subsequence $(x_n)_{n\in M}$ is weakly convergent to some $x_0\in X$. Moreover, if in addition $X$ has a Schauder basis then we may pass to a further subsequence $(x_n)_{n\in L}$ which is approximated by a sequence of the form $(\widetilde{x}_n)_{n\in L}$ such that $(\widetilde{x}_n)_{n\in L}$ also weakly converges to $x_0$ and $(\widetilde{x}_n-x_0)_{n\in L}$ is a block sequence of $X$. The main aim of this section is to show that, for every $k\geq2$, the $k$-sequences in Banach spaces satisfy similar properties.

A $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$, of a Banach space $X$ will be called weakly relatively compact if $\overline{\{x_s: s\in[{\mathbb{N}}]^k\}}^{w}$ is a weakly compact subset of $X$.

Since the weak topology on every separable weakly compact subset of a Banach space is metrizable, by Propositions \[subordinating yields convergence\] and \[Create subordinated\] we have the following.

\[cor for subordinating\] Let $X$ be a Banach space and $k\in{\mathbb{N}}$. Then we have the following.

1.  Every subordinated $k$-sequence in $(X,w)$ is weakly convergent.

2.  Every weakly relatively compact $k$-sequence in $X$ contains a subordinated $k$-subsequence.

To describe the regularity properties of weakly relatively compact $k$-sequences in a Banach space $X$ with Schauder basis we will need the next two definitions. The first is a natural extension of the notion of block (resp. disjointly supported) sequences of $X$.

Let $X$ be a Banach space with a Schauder basis and $k\in{\mathbb{N}}$. Let also $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $M\in[{\mathbb{N}}]^\infty$. We will say that the $k$-subsequence $(x_s)_{s\in[M]^k}$ is plegma block (resp. plegma disjointly supported) if for all plegma pairs $(s_1,s_2)$ in $[M]^k$ we have $\text{supp}(x_{s_1})<\text{supp}(x_{s_2})$ (resp. $\text{supp}(x_{s_1})\cap\text{supp}(x_{s_2})=\emptyset$).

\[Def of plegma supported\] Let $X$ a Banach space with a Schauder basis, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$. Also let $L\in
[{\mathbb{N}}]^\infty$ and $(y_t)_{t\in[L]^{\leq k}}$ be a family of vectors in $X$. We will say that $(y_t)_{t\in[L]^{\leq k}}$ is a canonical tree decomposition of $(x_s)_{s\in[L]^k}$ (or $(x_s)_{s\in[L]^k}$ admits $(y_t)_{t\in[L]^{\leq k}}$ as a canonical tree decomposition) if the following are satisfied.

1.  For every $s\in[L]^k$, $\displaystyle x_s=\sum_{j=0}^{k}y_{s|j}=
    y_\emptyset+\sum_{j=1}^{k}y_{s|j}$.

2.  For every $t\in[L]^{\leq k}\setminus\{\emptyset\}$, $\text{supp}(y_{t})$ is finite.

3.  For every $s\in[L]^k$ and $1\leq j_1<j_2\leq k$, $\text{supp}(y_{s|j_1})
    <\text{supp}(y_{s|j_2})$.

4.  For every $(s_1,s_2)\in\textit{Plm}_2([L]^k)$ and $1\leq j_1\leq j_2\leq k$, we have$$\text{supp}(y_{s_1|j_1}) <\text{supp}(y_{s_2|j_2})$$

5.  For every $(s_1,s_2)\in\textit{Plm}_2([L]^k)$ and $1\leq j_1<j_2\leq k$, we have $$\text{supp}(y_{s_2|j_1}) <\text{supp}(y_{s_1|j_2})$$

The next proposition gathers some basic properties of the $k$-sequences which admit canonical tree decomposition. Its proof is straightforward.

\[trocan\] Let $X$ a Banach space with a Schauder basis, $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $L\in
[{\mathbb{N}}]^\infty$. Assume that $(x_s)_{s\in[L]^k}$ admits $(y_t)_{t\in[L]^{\leq k}}$ as a canonical tree decomposition. Then the following are satisfied.

1.  For every $N\in[L]^\infty$, the $k$-subsequence $(x_s)_{s\in[N]^k}$ admits $(y_t)_{t\in[N]^{\leq k}}$ as a canonical tree decomposition.

2.  For every $s\in [L]^k$, the sequence $(y_{s|j})_{j=1}^k$ is a block sequence in $X$.

3.  For every $1\leq j\leq k$, the sequence $(y_{s|j})_{s\in [L]^k}$ is a plegma block $k$-sequence in $X$.

4.  Setting $x'_s=x_s-y_\emptyset$, for all $s\in[L]^k$, $y'_\emptyset=0$ and $y'_t=y_t$, for all $t\in[L]^{\leq k}$ with $t\neq\emptyset$, we have that the $k$-subsequence $(x'_s)_{s\in[L]^k}$ is plegma disjointly supported and admits $(y'_t)_{t\in[L]^{\leq k}}$ as a canonical tree decomposition.

5.  For every $j\in\{1,..,k\}$ and $(s_i)_{i=1}^n\in\text{Plm}_n([L]^k)$, if $I$ is the interval of ${\mathbb{N}}$ with $\min I=\min\text{supp}(y_{s_1|j})$ and $\max I= \max
    \text{supp}(y_{s_n|j})$, then for every $1\leq i\leq n$, $I(x_{s_i}-y_\emptyset)=y_{s_i|j}$.

The following is the main result of this section.

\[canonical tree\] Let $X$ be a Banach space with Schauder basis, $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $({\varepsilon}_n)_{n}$ be a null sequence of positive reals. Assume that for some $M\in
[{\mathbb{N}}]^\infty$, $(x_s)_{s\in[M]^k}$ is subordinated with respect to the weak topology of $X$ and let $x_0$ be the weak limit of $(x_s)_{s\in[M]^k}$. Then there exist $L\in[M]^\infty$ and a $k$-subsequence $(\widetilde{x}_s)_{s\in[L]^{k}}$ in $X$ satisfying the following.

1.  $(\widetilde{x}_s)_{s\in[L]^k}$ admits a canonical tree decomposition $(y_t)_{t\in[L]^{\leq k}}$ with $y_\emptyset=x_0$.

2.  For every $s\in[L]^k$, $\|x_s-\widetilde{x}_s\|<{\varepsilon}_n$, where $\min
    s=L(n)$.

3.  $(\widetilde{x}_s)_{s\in[L]^k}$ is subordinated with respect to the weak topology of $X$. Moreover $x_0$ is the weak limit of $(\widetilde{x}_s)_{s\in[L]^k}$.

Without loss of generality, we may assume that $({\varepsilon}_n)_n$ is decreasing. We will first define a family $(y_t)_{t\in [M]^k}$ of finitely supported vectors in $X$ as follows. Let $\widehat{\varphi}:[M]^{\leq k}\to (X,w)$ be the continuous map witnessing that $(x_s)_{s\in[M]^k}$ is subordinated. For $t=\emptyset$, we set $y_\emptyset=
\widehat{\varphi}(\emptyset)=x_0$. For $t\in[M]^{\leq k}\setminus\{\emptyset\}$, let $w_{t}=\widehat{\varphi}(t)-\widehat{\varphi}(t\setminus\{\max
t\})$. Notice that the sequence $(w_{t\cup\{m\}})_{m\in M}$ is weakly null, for all $t\in[M]^{< k}$. Hence, by a sliding hump argument, we may choose a family $\big\{I_t: t\in [M]^{\leq
k}\setminus\{\emptyset\}\big\}$ of finite intervals of ${\mathbb{N}}$ satisfying the following properties.

1.  For every $t\in [M]^{\leq k}$, with $t\neq \emptyset$, we have that $\|w_t-y_t\|=\|I_t^c(w_t)\|<{\varepsilon}_n/k$, where $M(n)=\max t$.

2.  For every $t\in [M]^{< k}$, $ \min
    I_{t\cup\{m\}}\stackrel{ m\in M}{\longrightarrow}\infty$.

Now for every $t\in [M]^{\leq k}\setminus \{\emptyset\}$, we set ${y}_t=I_t(w_t)$ and the definition of the family $(y_t)_{t\in
[M]^k}$ is completed. Also, for every $s\in[M]^k$, we set $\widetilde{x}_s=\sum_{t\sqsubseteq s}y_t$.

We claim that there exists $L\in [M]^\infty$ such that $(y_t)_{t\in
[L]^k}$ is a canonical tree decomposition of $(\widetilde{x}_s)_{s\in [L]^k}$. Indeed, using (P2) and Ramsey's theorem, there exists $M_1\in [M]^\infty$ such that for every $s\in[M_1]^k$ and $1\leq j_1<j_2\leq k$, $\text{supp}(y_{s|j_1})
<\text{supp}(y_{s|j_2})$. Using again (P2) and Theorem \[ramseyforplegma\], we find $M_2\in [M_1]^\infty$ such that for every $(s_1,s_2)\in\textit{Plm}_2([M_2]^k)$ and $1\leq j_1\leq
j_2\leq k$, $\text{supp}(y_{s_1|j_1}) <\text{supp}(y_{s_2|j_2})$, while for every $1\leq j_1<j_2\leq k$, $\text{supp}(y_{s_2|j_1})
<\text{supp}(y_{s_1|j_2})$. We set $L=M_2$. By the above, we have that all conditions (i)-(v) of Definition \[Def of plegma supported\] are fulfilled and therefore $(y_t)_{t\in [L]^{\leq k}}$ is a canonical tree decomposition of $(\widetilde{x}_s)_{s\in
[L]^k}$ and the proof of the claim is complete.

Notice that $x_s-\widetilde{x}_s=\sum_{j=1}^k (w_s|j-y_s|j)$, for all $s\in [L]^k$. Hence by (P1) and since $({\varepsilon}_n)_n$ is decreasing, we get that $\|x_s-\widetilde{x}_s\|\leq {\varepsilon}_n$, where $L(n)=\min s$. It remains to show that $(\widetilde{x}_s)_{s\in[L]^k}$ is subordinated. To this end, let $\widetilde{\varphi}:[L]^{\leq k}\to
X$ defined by $\widetilde{\varphi}(t)=\sum_{u\sqsubseteq t}y_u$, for all $t\in[L]^{\leq k}$. Clearly $\widetilde{\varphi}(\emptyset)=y_{\emptyset}=\widehat{\varphi}(\emptyset)$ and $\widetilde{x}_s=\widetilde{\varphi}(s)$, for all $s\in[L]^k$. To show that $\widetilde{\varphi}$ is continuous let $(t_n)_{n}$ be a sequence in $[L]^{\leq k}$ and $t\in [L]^{\leq k}$ such that $(t_n)_{n}$ converges to $t$. Setting $\max t_n= M(k_n)$, we may assume that $k_n\to\infty$. Then $$\begin{split}
  \|(\widehat{\varphi}(t_n)-\widehat{\varphi}(t))-(\widetilde{\varphi}(t_n)-\widetilde{\varphi}(t))\|\leq\sum_{t\sqsubset u\sqsubseteq
  t_n}\|w_u-y_u\|\leq {\varepsilon}_{k_n}\;\substack{\longrightarrow\\n\to\infty}\;0
\end{split}$$ Since $\widehat{\varphi}(t_n)\stackrel{w}{\to}\widehat{\varphi}(t)$, we get that $\widetilde{\varphi}(t_n)\stackrel{w}{\to}\widetilde{\varphi}(t)$ and the proof is completed.

Let $X$ be a Banach space and $k\in{\mathbb{N}}$. By $\mathcal{SM}_k^{wrc}(X)$ we will denote the set of all spreading sequences $(e_n)_n$ such that there exists a weakly relatively compact $k$-sequence of $X$ which generates $(e_n)_n$ as a $k$-spreading model. Notice that $\mathcal{SM}_k^{wrc}(X)=\mathcal{SM}_k(X)$, for every reflexive space $X$ and $k\in{\mathbb{N}}$.

\[cor canonical tree with spr mod\] Let $X$ be a Banach space with Schauder basis and $k\in{\mathbb{N}}$. Then every $(e_n)_{n}\in\mathcal{SM}_k^{wrc}(X)$ is generated by a $k$-sequence in $X$ which is subordinated with respect to the weak topology and admits a canonical tree decomposition.

Let $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a weakly relatively compact $k$-sequence in $X$ which generates a $k$-spreading model $(e_n)_{n}$. By Proposition \[cor for subordinating\], there exists $M\in[{\mathbb{N}}]^\infty$ such that $(x_s)_{s\in[M]^k}$ is subordinated. By Theorem \[canonical tree\], there exists $L\in [M]^\infty$ and a subordinated sequence $(\widetilde{x_s})_{s\in[L]^k}$ in $X$ which admits a canonical tree decomposition such that $\|x_s-\widetilde{x}_s\|<1/n$, for every $s\in [L]^k$ with $\min
s=L(n)$. Hence there is $N\in [L]^\infty$ such that $(\widetilde{x_s})_{s\in[N]^k}$ also generates $(e_n)_n$ as a $k$-spreading model. Setting $z_s=\widetilde{x}_{N(s)}$, for all $s\in[{\mathbb{N}}]^k$, we have that $(z_s)_{s\in [{\mathbb{N}}]^k}$ is as desired.

Norm properties of spreading models {#s5}
===================================

In this section we provide conditions for $k$-sequences to admit unconditional, singular or trivial spreading models. Our main interest concerns subordinated $k$-sequences with respect to the weak topology.

Unconditional spreading models
------------------------------

As is well known every spreading model generated by a seminormalized weakly null sequence is an $1$-unconditional spreading sequence. In this subsection we give an extension of this result for subordinated seminormalized weakly null $k$-sequences.

\[Lemma finding convex means\] Let $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in a Banach space $X$. Suppose that $(x_s)_{s\in[{\mathbb{N}}]^k}$ is subordinated and let $\widehat{\varphi}:[{\mathbb{N}}]^{\leq k}\to (X,w)$ be the continuous map witnessing this. Let ${\varepsilon}>0$, $M\in
[{\mathbb{N}}]^\infty$ and $ n\in{\mathbb{N}}$. Then for every $p\in\{1,...,n\}$ there exists a finite subset $G$ of $[M]^k$ such that the following are satisfied.

1.  There exists a convex combination $x=\sum_{s\in
    G}\mu_sx_s$ of $(x_s)_{s\in G}$ such that $\|\widehat{\varphi}(\emptyset)-x\|<{\varepsilon}$.

2.  For every $ 1\leq i\leq n$ with $i\neq p$, there exists $s_i\in [M]^k$ such that for every $s_p\in G$, the family $(s_i)_{i=1}^n$ is a plegma family in $[M]^k$.

For $k=1$, the result follows by Mazur's theorem. We proceed by induction on $k\in{\mathbb{N}}$. Assume that the lemma is true for some $k\in{\mathbb{N}}$. We fix a subordinated $(k+1)$-sequence $(x_s)_{s\in[{\mathbb{N}}]^{k+1}}$ in $X$, $M\in[{\mathbb{N}}]^\infty$, $ n\in {\mathbb{N}}$, ${\varepsilon}>0$ and $p\in\{1,...,n\}$.

Let $(x_t)_{t\in[M]^k}$ defined by $x_t=\widehat{\varphi}(t)$, for all $t\in [M]^k$. By our inductive assumption, there exists a finite subset $F$ of $[M]^k$ satisfying the following.

1.  There exists a convex combination $\sum_{t\in
    F}\mu_tx_t$ of $(x_t)_{t\in F}$ such that $$\label{hg}\Big\|\widehat{\varphi}(\emptyset)-\sum_{t\in
    F}\mu_tx_t\Big\|<{\varepsilon}/2$$

2.  For every $ 1\leq
    i\leq n$ with $i\neq p$, there exists $t_i\in [M]^k$ such that for every $t_p\in F$, $(t_i)_{i=1}^n$ is a plegma family in $[M]^k$.

For notational simplicity we assume that $1<p<n$ (the proof for $p\in\{1,n\}$ is similar). Pick $m_1<\ldots<m_{p-1}$ in $M$ with $t_n(k)< m_1$ and set $s_i=t_i\cup \{m_i\}$, for all $i=1,\ldots,p-1$. Also let $M'=\{m\in
M:m>m_{p-1}\}$. Since $\widehat{\varphi}$ is continuous, we have that $(x_{t\cup\{m\}})_{m\in M'}\stackrel{w}{\to}x_t$, for every $t\in F$. Hence by Mazur's theorem, for every $t\in F$, there exists a finite subset $G_t$ of $M'$ such that $$\label{hgg}\Big\|x_t-\sum_{m\in
G_t}\mu^t_mx_{t\cup\{m\}}\Big\|<{\varepsilon}/2$$ for some convex combination $\sum_{m\in G_t}\mu^t_m
x_{t\cup\{m\}}$ of $(x_{t\cup\{m\}})_{m\in G_t}$. We set $$G=\{t\cup\{m\}:t\in F\;\text{and}\;m\in
G_t\}$$ Finally, pick $m_{p+1}<...<m_n$ in $M$ with $\max\{m:m\in\bigcup_{t\in F} G_t\}<m_{p+1}$ and let $s_i=t_i\cup\{m_i\}$, for all $i=p+1,\ldots,n$.

It is easy to check that every $(s_i)_{i=1}^n$ with $s_p\in G$, is a plegma family in $[M]^{k+1}$. It remains to show that condition (i) of the lemma is also satisfied. To this end, let $\mu_s=\mu_t\mu^t_m$, for every $s=t\cup\{m\}\in G$, where $\max t<m$. Notice that $$\sum_{s\in G}\mu_s=\sum_{t\in F}\mu_t\sum_{m\in G_t}\mu^t_m=\sum_{t\in F}\mu_t=1$$ and therefore $\sum_{s\in G}\mu_s x_s$ is a convex combination of $(x_s)_{s\in G}$. Moreover, we have $$\begin{split}
\Big\|\widehat{\varphi}(\emptyset)-\sum_{s\in G}&\mu_sx_s\Big\|=
\Big\|\widehat{\varphi}(\emptyset)-\sum_{t\in F}\mu_t\sum_{m\in G_t}\mu^t_m x_{t\cup\{m\}}\Big\|\\
&\leq\Big\|\widehat{\varphi}(\emptyset)-\sum_{t\in
G'}\mu'_tx_t\Big\|+ \sum_{t\in F}\mu_t\cdot\Big\|x_t-\sum_{m\in
G_t}\mu^t_mx_{t\cup\{m\}}\Big\|\stackrel{(\ref{hg}),
(\ref{hgg})}{<}{\varepsilon}\end{split}$$ and the proof is complete.

\[unconditional spreading model\] Let $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in a Banach space $X$. Suppose that $(x_s)_{s\in[{\mathbb{N}}]^k}$ is seminormalized, subordinated (with respect to the weak topology of $X$) and weakly null. Then every $k$-spreading model of $(x_s)_{s\in[{\mathbb{N}}]^k}$ is $1$-unconditional.

Let $(e_n)_n$ be a spreading model of $(x_s)_{s\in[{\mathbb{N}}]^k}$. Lemma \[Lemma finding convex means\] and the averaging technique used for the proof of the corresponding result in the case of the classical spreading models (see @BL Proposition I.5.1) yield that for every $n\in{\mathbb{N}}$, $1\leq
p\leq n$, $a_1,\ldots,a_n\in [-1,1]$ and ${\varepsilon}>0$, we have $$\Big{\|}\sum_{\substack{i=1\\i\neq
p}}^na_ie_i\Big{\|}_*\leq\Big{\|}\sum_{i=1}^na_ie_i\Big{\|}_*+\varepsilon$$ Since the above inequality holds for every ${\varepsilon}>0$, we have that $$\label{eq13}\Big{\|}\sum_{\substack{i=1\\i\neq
p}}^na_ie_i\Big{\|}_*\leq\Big{\|}\sum_{i=1}^na_ie_i\Big{\|}_*$$ for all $n\in{\mathbb{N}}$, $1\leq p\leq n$ and $a_1,\ldots,a_n\in [-1,1]$. Since $(x_s)_{s\in[{\mathbb{N}}]^k}$ is seminormalized, we have that $\|e_1\|_*>0$. By (\[eq13\]) we get that $\|e_1-e_2\|_*>0$. By Proposition \[sing\], we get that $(e_n)_n$ is non trivial. An iterated use of (\[eq13\]) completes the proof.

We close this subsection by giving an example showing that for $k\geq 2$ the assumption in Theorem \[unconditional spreading model\] that the $k$-sequence is subordinated is necessary. More precisely, for every $k\geq 2$, there exist seminormalized weakly null $k$-sequences which generate conditional Schauder basic spreading models.

For simplicity we state the example for $k=2$. Let $(e_n)_n$ be the usual basis of $c_0$ and $(x_s)_{s\in[{\mathbb{N}}]^2}$ be the $2$-sequence in $c_0$, defined by $x_s=\sum_{n=\min s}^{ \max s}e_n$, for all $s\in[{\mathbb{N}}]^2$. Clearly, $(x_s)_{s\in[{\mathbb{N}}]^2}$ is a normalized weakly null $2$-sequence. It is easy to check that for all $l\in{\mathbb{N}}$, $a_1,\ldots,a_l\in{\mathbb{R}}$ and $(s_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^2)$, we have $$\Big\|\sum_{j=1}^la_jx_{s_j}\Big\|=\max\Big(\max_{1\leq k\leq
l}\Big|\sum_{j=1}^ka_j\Big|,\max_{1\leq k\leq
\l}\Big|\sum_{j=k}^la_j\Big|\Big)$$ Therefore every spreading model of $(x_s)_{s\in[{\mathbb{N}}]^2}$, is equivalent to the summing basis.

Singular and trivial spreading models
-------------------------------------

The results of this subsection concern the $k$-spreading models generated by subordinated $k$-sequences which are not weakly null.

\[triv-ell\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $x_0\in X$. Let $x'_s=x_s-x_0$, for all $s\in[{\mathbb{N}}]^k$ and assume that that $(x_s)_{s\in[{\mathbb{N}}]^k}$ and $(x'_s)_{s\in[{\mathbb{N}}]^k}$ generate $k$-spreading models $(e_n)_n$ and $(\widetilde{e}_n)_n$ respectively. Then the following hold.

1.  $\|\sum_{i=1}^na_ie_{_i}\|=\|\sum_{i=1}^na_i\widetilde{e}_{_i}\|$, for every $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ with $\sum_{i=1}^na_i=0$.

2.  The sequence $(e_n)_n$ is trivial if and only if $(\widetilde{e}_n)_n$ is trivial.

3.  The sequence $(e_n)_n$ is equivalent to the usual basis of $\ell^1$ if and only if $(\widetilde{e}_n)_n$ is equivalent to the usual basis of $\ell^1$.

\(a) Notice that for every $n\in{\mathbb{N}}$, $s_1,...,s_n$ in $[{\mathbb{N}}]^k$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ with $\sum_{i=1}^na_i=0$, we have $\sum_{i=1}^na_ix_{s_i}=\sum_{i=1}^na_ix'_{s_i}$. Since $(e_n)_n$ and $(\widetilde{e}_n)_n$ are generated by $(x_s)_{s\in [{\mathbb{N}}]^k}$ and $(x_s)_{s\in [{\mathbb{N}}]^k}$ the result follows.\
(b) It follows by assertion (a) and Proposition \[sing\].\
(c) We fix ${\varepsilon}>0$. If $(\widetilde{e}_n)_n$ is not equivalent to the usual basis of $\ell^1$ then there exist $n\in{\mathbb{N}}$ and $a'_1,\ldots,a'_n\in{\mathbb{R}}$ such that $\sum_{i=1}^n|a'_i|=1$ and $\|\sum_{i=1}^na'_i\widetilde{e}_i\|<{\varepsilon}$. Setting $a_i=a'_i/2$ and $a_{n+i}=-a'_i/2$, for all $1\leq i\leq n$, we have $\sum_{i=1}^{2n}a_i=0$ and therefore, $\|\sum_{i=1}^{2n}a_ie_i\|=\|\sum_{i=1}^{2n}a_i\widetilde{e}_i\|<{\varepsilon}$. Since $\sum_{i=1}^{2n}|a_i|=1$, $(e_n)_n$ is also not equivalent to the usual basis of $\ell^1$.

\[nb\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a subordinated $k$-sequence in $X$. Also let $x'_s=x_s-x_0$, for every $s\in[{\mathbb{N}}]^k$, where $x_0$ is the weak limit of $(x_s)_{s\in[{\mathbb{N}}]^k}$. Assume that for some $M\in [{\mathbb{N}}]^\infty$ the $k$-subsequence $(x_s)_{s\in[M]^k}$ generates a non trivial $k$-spreading model $(e_n)_n$. If $x_0\neq 0$, then exactly one of the following holds.

1.  The sequence $(e_n)_n$ as well as every spreading model of $(x'_s)_{s\in [M]^k}$ is equivalent to the usual basis of $\ell^1$.

2.  The sequence $(e_n)_n$ is singular and if $e_n=e'_n+e$ is its natural decomposition then $(e'_n)_n$ is the unique $k$-spreading model of $(x'_s)_{s\in[M]^k}$ and $\|e\|=\|x_0\|$.

Let $(\widetilde{e}_n)_n$ be a $k$-spreading model of $(x'_s)_{s\in[M]^k}$. If $(e_n)_n$ is equivalent to the usual basis of $\ell^1$ then by Lemma \[triv-ell\], we have that the same holds for $(\widetilde{e}_n)_n$ and hence (i) is satisfied.

Assume for the following that $(e_n)_n$ is not equivalent to the usual basis of $\ell^1$. Since it is also non trivial, by Lemma \[triv-ell\], we have that $(\widetilde{e}_n)_n$ is non trivial and not equivalent to the $\ell^1$-basis. Let $L\in
[M]^\infty$ such that $(x'_s)_{s\in[L]^k}$ generates $(\widetilde{e}_n)_n$. Since $(\widetilde{e}_n)_n$ is non trivial, it is easy to see that $(x'_s)_{s\in[L]^k}$ is seminormalized. Also notice that $(x'_s)_{s\in[M]^k}$ is subordinated and weakly null. Therefore by Theorem \[unconditional spreading model\], $(\widetilde{e}_n)_n$ is 1-unconditional. Moreover, since $(\widetilde{e}_n)_n$ is not equivalent to the usual basis of $\ell^1$, by Proposition \[equiv forms for 1-subsymmetric weakly null\], we conclude that $(\widetilde{e}_n)_n$ is Cesàro summable to zero. Hence we have $$\label{tv} \lim_{n\to\infty}\Big\|\frac{1}{n}\sum_{j=1}^ne_j-\frac{1}{n}\sum_{j=n+1}^{2n}e_j\Big\|
=\lim_{n\to\infty}\Big\|\frac{1}{n}\sum_{j=1}^n\widetilde{e}_j-\frac{1}{n}\sum_{j=n+1}^{2n}\widetilde{e}_j\Big\|=0$$ Also it is easy to see that $$\label{fd}
\Big\|\frac{1}{n}\sum_{j=1}^ne_j\Big\|\to\|x_0\|>0$$ By (\[tv\]) and (\[fd\]), we get that $(e_n)_n$ is not Schauder basic, i.e. it is singular. Let $e_n=e'_n+e$ be the natural decomposition of $(e_n)_n$. By (\[fd\]) and the fact that $(e'_n)_n$ is Cesàro summable to zero, we have that $\|e\|=\|x_0\|$. To complete the proof it remains to show that $(\widetilde{e}_n)_n$ and $(e'_n)_n$ are isometrically equivalent. Indeed, we fix $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$. For every $p\in{\mathbb{N}}$, let $(s^p_{j})_{j=1}^{n+p}\in\textit{Plm}_p([L]^k)$ such that $s^p_1(1)\geq L(n+p)$. We also set $a=\sum_{j=1}^na_j$. Then we have $$\begin{split}
    \Big\|\sum_{j=1}^na_je'_j\Big\|&
    =\lim_{p\to\infty}\Big\|\sum_{j=1}^na_je'_j-\frac{a}{p}\sum_{j=n+1}^{n+p}e'_j\Big\|
    =\lim_{p\to\infty}\Big\|\sum_{j=1}^na_je_j-\frac{a}{p}\sum_{j=n+1}^{n+p}e_j\Big\|\\
    &=\lim_{p\to\infty}\Big\|\sum_{j=1}^na_jx_{s_j^p}-\frac{a}{p}\sum_{j=n+1}^{n+p}x_{s_j^p}\Big\|
    =\lim_{p\to\infty}\Big\|\sum_{j=1}^na_jx'_{s_j^p}-\frac{a}{p}\sum_{j=n+1}^{n+p}x'_{s_j^p}\Big\|\\
    &=\lim_{p\to\infty}\Big\|\sum_{j=1}^na_j\widetilde{e}_j-\frac{a}{p}\sum_{j=n+1}^{n+p}\widetilde{e}_j\Big\|
    =\Big\|\sum_{j=1}^na_j\widetilde{e}_j\Big\|
  \end{split}$$

By Remark \[properties of the natural decomposition\], Proposition \[cor for subordinating\] and Theorems \[unconditional spreading model\] and \[nb\], we derive the following.

\[cor singular wrc\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(e_n)_{n}\in\mathcal{SM}^{wrc}_k(X)$ non trivial. Then one of the following holds.

1.  The sequence $(e_n)_n$ is unconditional.

2.  The sequence $(e_n)_n$ is singular and if $e_n=e'_n+e$ is the natural decomposition of $(e_n)_n$ then $(e'_n)_{n}\in\mathcal{SM}^{wrc}_k(X)$, $(e'_n)_n$ is unconditional, weakly null and Cesàro summable to zero. Moreover, the spaces generated by $(e_n)_n$ and $(e'_n)_n$ are isomorphic.

The next theorem provides more information concerning the trivial $k$-spreading models. Since we shall not use this result in the sequel, we omit its proof.

\[Theorem equivalent forms for having norm on the spreading model\] Let $k\in{\mathbb{N}}$, $(x_s)_{s\in[{\mathbb{N}}]^k}$ be an $k$-sequence in a Banach space $X$ and $(E,\|\cdot\|_*)$ be an infinite dimensional seminormed linear space with Hamel basis $(e_n)_n$. Assume that for some $M\in[{\mathbb{N}}]^\infty$, the $k$-subsequence $(x_s)_{s\in
[M]^k}$ generates $(e_n)_n$ as an $k$-spreading model. Then the following are equivalent:

1.  The sequence $(e_n)_n$ is trivial.

2.  The seminorm $\|\cdot\|_*$ is not a norm on $E$.

3.  $(x_s)_{s\in [M]^k}$ contains a further norm Cauchy $k$-subsequence, i.e. there exists $L\in[M]^\infty$ such that for every ${\varepsilon}>0$ there exists $n_0\in{\mathbb{N}}$ satisfying that $\|x_s-x_t\|<{\varepsilon}$, for all $s,t\in[L]^k$ with $n_0\leq\min\{\min
    s,\min t\}$.

4.  There exists $x\in X$ such that every $k$-subsequence of $(x_s)_{s\in[M]^k}$ contains a further $k$-subsequence convergent to $x$.

Composition of the spreading models
===================================

In this section we study the composition property of the $k$-spreading models. Moreover we recall the definition of the $k$-iterated spreading models and we investigate their relation with the $k$-spreading models. We start with the following definition.

Let $X$ be a Banach space with a Schauder basis and $k\in{\mathbb{N}}$. Then a $k$-spreading model $(e_n)_n$ of $X$ will be called plegma block generated if there exists a $k$-sequence $(x_s)_{s\in
[{\mathbb{N}}]^k}$ which is plegma block and generates $(e_n)_n$ as a $k$-spreading model.

\[remnj\] By Lemma \[propxi\], we easily conclude that for $1\leq
k_1<k_2$, every plegma block generated $k_1$-spreading model is also a plegma block $k_2$-spreading model. Thus the plegma block $k$-spreading models of a Banach space $X$ with a Schauder basis form an increasing hierarchy.

\[composition thm\] Let $X$ be a Banach space, $k\in {\mathbb{N}}$ and $(e_n)_n\in
\mathcal{SM}_{k}(X)$ such that $(e_n)_n$ is a Schauder basic sequence. Let $E$ be the Banach space with Schauder basis the sequence $(e_n)_n$, $d\in{\mathbb{N}}$ and $(\widetilde{e}_n)_{n}$ be a plegma block generated $d$-spreading model of $E$. Then $(\widetilde{e}_n)_{n}\in\mathcal{SM}_{k+d}(X)$.

We fix a plegma block $d$-sequence $(y_t)_{t\in[{\mathbb{N}}]^d}$ in $E$ which generates $(\widetilde{e}_n)_n$ as a $d$-spreading model with respect to some null sequence $(\widetilde{\delta}_n)_n$ of positive reals. By Proposition \[remark on the definition of spreading model\], we may also choose a $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ in $X$ which generates $(e_n)_n$ as a $k$-spreading model with respect to the same sequence $(\widetilde{\delta}_n)_n$.

Since $(y_t)_{t\in [{\mathbb{N}}]^d}$ is finitely supported, setting for every $t\in[{\mathbb{N}}]^d$, $F_t=\text{supp}(y_t)$, $$y_t=\sum_{j=1}^{|F_t|}a^t_{F_t(j)}e_{F_t(j)}^{\;}$$ For every $v\in[{\mathbb{N}}]^{k+d}$, let $t_v$ (resp. $s_v$) be the unique element in $[{\mathbb{N}}]^d$ (resp. $[{\mathbb{N}}]^k$) such that $v=t_v\cup s_v$ and $t_v<s_v$. For every $v\in [{\mathbb{N}}]^{k+d}$ and $j\in\{1,...,|F_{t_u}|\}$, we set $$\label{pl}s_j^v=(s_v(1)+j-1,...,s_v(k)+j-1)$$ Notice that $(s_j^v)_{j=1}^{|F_{t_v}|}$ is a finite sequence in $[{\mathbb{N}}]^k$ with $s_1^v=s_v$.

We define a $(k+d)$-sequence $(z_v)_{v\in[{\mathbb{N}}]^{k+d}}$ in $X$, by setting $$z_v=\sum_{j=1}^{|F_{t_v}|}a^{t_v}_{F_{t_v}(j)}x_{s^v_j}$$ The proof will be completed once we show the following.

**Claim 1.** There exists $M\in[{\mathbb{N}}]^\infty$ such that $(z_v)_{v\in[M]^{k+d}}$ generates $(\widetilde{e}_n)_n$ as a $(k+d)$-spreading model.

*Proof of Claim 1*: For every $l\in{\mathbb{N}}$, we define a family $\mathcal{A}_l\subseteq \textit{Plm}_l([{\mathbb{N}}]^{k+d})$ as follows: $$\begin{split}\mathcal{A}_l=\Big{\{} (v_i)_{i=1}^l\in &\textit{Plm}_l([{\mathbb{N}}]^{k+d}):
  \; s_1^{v_1}(1)\geq \sum_{i=1}^{l}|F_{t_{v_i}}| \\
  &\text{and}\;\;
  (s_j^{v_1})_{j=1}^{|F_{t_{v_1}|}}\;^\frown \ldots \;^\frown (s_j^{v_l})_{j=1}^{|F_{t_{v_l}|}}\in
  \textit{Plm}_{\sum_{i=1}^{l}|F_{t_{v_i}}|}([{\mathbb{N}}]^{k})
   \Big\}\end{split}$$ Using (\[pl\]), the fact that for every $(v_i)_{i=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^{k+d})$, $(s_{v_i})_{i=1}^l\in \textit{Plm}_l([{\mathbb{N}}]^{k})$ and that $s_1^{v_i}=s_{v_i}$, for all $1\leq i\leq l$, it is easy to check that $\mathcal{A}_l\bigcap \textit{Plm}_l([L]^{k+d})\neq
\emptyset$, for every $l\in{\mathbb{N}}$ and $L\in [{\mathbb{N}}]^\infty$. Hence, an iterated use of Theorem \[ramseyforplegma\], yields an $L\in
[{\mathbb{N}}]^\infty$ such that $(v_i)_{i=1}^l\in\mathcal{A}_l$, for every $(v_i)_{i=1}^l\in\textit{Plm}_l([L]^{k+d})$, with $v_1(1)\geq
L(l)$.

We fix $l\in{\mathbb{N}}$, $(v_i)_{i=1}^l\in\textit{Plm}_l([L]^{k+d})$ with $v_1(1)\geq L(l)$ and $a_1,\ldots,a_l\in[-1,1]$. Notice that $$\begin{split}\label{eq10}
\Bigg| \Big\|\sum_{i=1}^l a_i z_{v_i} \Big\|-\Big\|\sum_{i=1}^l a_i \widetilde{e}_i \Big\| \Bigg|
\leq& \Bigg|\Big\|\sum_{i=1}^l a_i z_{v_i}   \Big\|-\Big\|\sum_{i=1}^l a_i y_{t_{v_i}}  \Big\| \Bigg|\\
&+ \Bigg| \Big\|\sum_{i=1}^l a_i y_{t_{v_i}}  \Big\|- \Big\|\sum_{i=1}^l a_i \widetilde{e}_i \Big\| \Bigg|
\end{split}$$ Also observe that $(t_{v_i})_{i=1}^l\textit{Plm}_l([L]^{d\widehat{}})$ and $t_{v_1}(1)=v_1(1)\geq L(l)\geq l$. Hence, $$\label{eq11}
\Bigg| \Big\|\sum_{i=1}^l a_i y_{t_{v_i}}  \Big\|-\Big\| \sum_{i=1}^l a_i \widetilde{e}_i  \Big\|\Bigg|<\widetilde{\delta}_l$$ Also, $s_1^{v_1}(1)\geq \sum_{i=1}^l |F_{t_{v_i}}|$ and $F_{t_{v_1}}<...<F_{t_{v_l}}$. Therefore, $$\begin{split}\label{eq12}
\Bigg|\Big\|\sum_{i=1}^l a_i z_{v_i}   \Big\|-\Big\|\sum_{i=1}^l a_i y_{t_{v_i}}  \Big\| \Bigg|=&
\Bigg|\Big\|\sum_{i=1}^l\sum_{j=1}^{l_{t_{v_i}}} a_ia_{F_{t_{v_i}}(j)}^{t_{v_i}} x_{s^{v_i}_j}  \Big\|\\
&-\Big\|\sum_{i=1}^l\sum_{j=1}^{l_{t_{v_i}}} a_ia_{F_{t_{v_i}}(j)}^{t_{v_i}} e_{F_{t_{v_i}}(j)}^{\;}  \Big\|  \Bigg|<2CK\widetilde{\delta}_l
\end{split}$$ where $C$ is the basis constant of $(e_n)_n$ and $K=\sup\{\|y_t\|:t\in[L]^k\}$.

By (\[eq10\]), (\[eq11\]) and (\[eq12\]), we obtain that for every $l\in{\mathbb{N}}$, $(v_i)_{i=1}^l\in\textit{Plm}_l([L]^{k})$ with $v_1(1)\geq L(l)$ and $a_1,\ldots,a_l\in[-1,1]$, we have $$\Bigg| \Big\|\sum_{i=1}^l a_i z_{v_i} \Big\|-\Big\|\sum_{i=1}^l a_i
\widetilde{e}_i \Big\| \Bigg|<\delta_l$$ where $\delta_l=(1+2CK)\widetilde{\delta}_l$. By Lemma \[old defn yields new\], there exists $M\in[L]^\infty$, such that $(z_v)_{v\in[M]^{k}}$ generates $(\widetilde{e}_n)_{n}$ as a $k$-spreading model and the proof of the claim as well as of Theorem \[composition thm\] is complete.

Let $X$ be a Banach space and $Y$ be either $\ell^p$ for some $p\in[1,\infty)$ or $c_0$. Also let $k\in{\mathbb{N}}$, $(e_n)_n\in\mathcal{SM}^{wrc}_k(X)$ be non trivial and $E$ be the Banach space generated by $(e_n)_n$. Suppose that $E$ contains an isomorphic copy of $Y$. Then $\mathcal{SM}_{k+1}(X)$ contains a sequence equivalent to the usual basis of $Y$.

First assume that that $(e_n)_n$ is Schauder basic. Notice that $E$ contains a block sequence $(y_n)_n$ equivalent to the usual basis of $Y$. It is easy to see that $(y_n)_n$ admits a spreading model $(\widetilde{e}_n)_n$ equivalent to the usual basis of $Y$. By Theorem \[composition thm\] we have that $(\widetilde{e}_n)_n\in\mathcal{SM}_{k+1}(X)$.

Assume now that $(e_n)_n$ is not Schauder basic. Since $(e_n)_n$ is non trivial, we have that $(e_n)_n$ is singular. Let $e_n=e'_n+e$ be its natural decomposition and $E'$ the space generated by $(e'_n)_n$. By Remark \[properties of the natural decomposition\] we have that $E$ and $E'$ are isomorphic and therefore $E'$ contains an isomorphic copy of $Y$. By Corollary \[cor singular wrc\] we have that $(e'_n)_n\in\mathcal{SM}_{k+1}(X)$. Since $(e'_n)_n$ is unconditional, the result follows as in the first case.

The $k$-iterated spreading models
---------------------------------

In this subsection we define the $k$-iterated spreading models of a Banach space $X$ which although they have not been named, have been appeared in @BM and @O-S. We also study their relation with the $k$-spreading models.

The $k$-iterated spreading models of a Banach space $X$ are inductively defined as follows. The 1-iterated are the non trivial spreading models of $X$. Assume that for some $k\in{\mathbb{N}}$ the $k$-iterated spreading models of $X$ have been defined. Then the $(k+1)$-iterated spreading models are the non trivial spreading models of the spaces generated by the $k$-iterated spreading models.

Notice that the class of the $k$-iterated spading models of a Banach space $X$ is contained in the one of the $(k+1)$-iterated spreading models. In the sequel we provide a sufficient condition ensuring that the $k$-iterated spreading models of a Banach space $X$ are up to isomorphism contained in $\mathcal{SM}_k(X)$. To this end we need the following lemma.

\[iter\_inter\] Let $X$ be a Banach space and $k\in{\mathbb{N}}$. Let $(e^0_n)_n$ be a Schauder basic $k$-spreading model of $X$, $E_0$ be the space generated by $(e^0_n)_n$, $(e_n)_n$ be a non trivial spreading model of $E_0$ and $E$ be the space generated by $(e_n)_n$. If $E_0$ is reflexive then there exists an unconditional $(k+1)$-spreading model of $X$ generating a space isomorphic to $E$.

Let $(x_n)_n$ be a sequence in $E_0$ generating $(e_n)_n$ as a spreading model. Since $E_0$ is reflexive, we may assume that $(x_n)_n$ is weakly convergent to some $x_0\in E_0$. If $x_0=0$, then $(e_n)_n$ is unconditional and it is generated by a block sequence in $E_0$, while if $(e_n)_n$ is equivalent to the usual basis of $\ell^1$ then $E_0$ contains a block sequence generating an $\ell^1$ spreading model. Therefore, in both cases the result follows by Theorem \[composition thm\]. Assume that $x_0\neq0$ and $(e_n)_n $ is not equivalent to the usual basis of $\ell^1$. Let $x'_n=x_n-x_0$, for all $n\in{\mathbb{N}}$. By Theorem \[nb\], we have that $(e_n)_n$ is singular and $(e'_n)_n$ is the unique spreading model of $(x'_n)_n$, where $e_n=e'_n+e$ is the natural decomposition of $(e_n)_n$. Since $(x'_n)_n$ is weakly null, we have that $(e'_n)_n$ is generated by a block sequence in $E_0$ as a spreading model. Hence, by Theorem \[composition thm\], the sequence $(e'_n)_n$ is a $(k+1)$-spreading model of $X$. Moreover, by Remark \[properties of the natural decomposition\], $(e'_n)_n$ is unconditional and the space $E'$ generated by $(e'_n)_n$ is isomorphic to $E$.

\[iterated\_lemma\] Let $X$ be a reflexive space and $k\in{\mathbb{N}}$ such that every space generated by a $k$-iterated spreading model of $X$ is reflexive. Then every space generated by a $(k+1)$-iterated spreading model of $X$ is isomorphic to the space generated by an unconditional $(k+1)$-spreading model of $X$.

We first treat the case $k=1$. So assume that $X$ as well as every space generated by a spreading model of $X$ is reflexive. Let $(\widetilde{e}_n)_n$ be a $2$-iterated spreading model of $X$ and $\widetilde{E}$ be the space generated by $(\widetilde{e}_n)_n$. Also let $\widetilde{E}_0$ be the space generated by a spreading model of $X$ such that $(\widetilde{e}_n)_n$ is a spreading model of $\widetilde{E}_0$. Since $X$ is reflexive, by Corollary \[cor singular wrc\], we conclude that $\widetilde{E}_0$ is isomorphic to a space $E_0$, generated by an unconditional spreading model of $X$. Moreover, by our assumption $E_0$ is also reflexive. Summarizing, the space $E_0$ is reflexive, it has a Schauder basis which is a spreading model of $X$ and it is isomorphic to $\widetilde{E}_0$. Therefore, $E_0$ admits a spreading model $(e_n)_n$ equivalent to $(\widetilde{e}_n)_n$. Let $E$ be the space generated by $(e_n)_n$. By Lemma \[iter\_inter\], there exists an unconditional $2$-spreading model of $X$ generating a space isomorphic to $E$. Since $E$ is isomorphic to $\widetilde{E}$ the proof of the proposition for $ k=1$ is completed.

We proceed by induction. Assume that the proposition holds for some $k\in{\mathbb{N}}$ and let $X$ be a reflexive space such that every space generated by a $(k+1)$-iterated spreading model of $X$ is reflexive. Let $(\widetilde{e}_n)_n$ be a $(k+2)$-iterated spreading model of $X$ and $\widetilde{E}$ be the space that it generates. Let $\widetilde{E}_0$ be the space generated by a $(k+1)$-iterated spreading model of $X$ admitting $(\widetilde{e}_n)_n$ as a spreading model. Since the $k$-iterated spreading models of $X$ are included in the $(k+1)$-iterated ones, we have that the spaces generated by the $k$-iterated spreading models of $X$ are reflexive. Hence, by our assumption that the proposition holds for the positive integer $k$, we have that $\widetilde{E}_0$ is isomorphic to some space $E_0$ generated by an unconditional $(k+1)$-spreading model of $X$. Therefore, $E_0$ is reflexive, it is generated by a Schauder basic $(k+1)$-spreading model of $X$ and admits a spreading model $(e_n)_n$ equivalent to $(\widetilde{e}_n)_n$. Let $E$ be the space generated by $(e_n)_n$. By Lemma \[iter\_inter\], there exists an unconditional $k+2$-spreading model of $X$ generating a space isomorphic to $E$. Since $E$ is isomorphic to $\widetilde{E}$ the proof of is completed.

\[qwqwe\] Let $X$ be a reflexive space such that for every $k\in{\mathbb{N}}$, every space generated by an unconditional $k$-spreading model of $X$ is reflexive. Then for every $k\in{\mathbb{N}}$, every space generated by a $k$-iterated spreading model of $X$ is isomorphic to the space generated by an unconditional $k$-spreading model of $X$.

By Corollary \[cor singular wrc\] we have that every space generated by a spreading model of $X$ is isomorphic to the space generated by an unconditional spreading model of $X$ and therefore it is reflexive. The proof is carried out by induction and using Proposition \[iterated\_lemma\].

As it is well known, see @BL, every non trivial spreading model of $c_0$ generates a space isomorphic to $c_0$. This easily implies that every $k$-iterated spreading model of $c_0$ generates a space isomorphic to $c_0$. On the other hand, as we will see in Section , the class of the $2$-spreading models of $c_0$ includes all spreading bimonote Schauder basic sequences yielding the existence of 2-spreading models which are not 2-iterated ones.

H.P. Rosenthal had asked whether every 2-iterated spreading model of a Banach space $X$ is actually a classical one. In @BM a Banach space $X$ has been constructed not admitting $\ell^1$ as a spreading model, while there is a spreading model generating a space which contains $\ell^1$. Thus $\ell^1$ occurs as 2-iterated spreading model but not as a classical one. A more striking result (see @AOTS) asserts the existence of a Banach space $X$ not admitting $\ell^1$ as a spreading model but $\ell^1$ is isomorphic to a subspace of every space generated generated by a non trivial spreading model of $X$. It remains open if for every $k\in{\mathbb{N}}$ there exists a Banach space $X_{k+1}$ such that the class of $(k+1)$-iterated spreading models strictly includes the corresponding one of $k$-iterated.

$k$-spreading models equivalent to the $\ell^1$ basis {#chapter 9}
=====================================================

In this section we study the properties of the $k$-spreading models equivalent to the usual basis of $\ell^1$.

Splitting spreading sequences equivalent to the $\ell^1$ basis
--------------------------------------------------------------

In this subsection we present some stability properties of spreading sequences in seminormed linear spaces which are actually related to the non distortion of $\ell^1$ (c.f. @J2).

Let $(e_n)_{n}$ be a spreading sequence in a seminormed linear space $(E, \|\cdot\|_*)$ and $c>0$. We say that $(e_n)_n$ *admits a lower $\ell^1$-estimate of constant* $c$, if for every $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$, we have $c\sum_{i=1}^n
|a_i|\leq \big\|\sum_{i=1}^n a_i e_i\big \|_*$.

\[propsplitl1\] Let $(E,\|\cdot\|_\circ),(E_1,\|\cdot\|_*),(E_2,\|\cdot\|_{**})$ be seminormed linear spaces and $(e_n)_{n},(e_n^1)_{n}$ and $(e_n^2)_{n}$ be spreading sequences in $E, E_1$ and $E_2$ respectively. Assume that for every $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$, we have $$\label{gt1}\Big\|\sum_{i=1}^na_ie_i\Big\|_\circ\leq
\Big\|\sum_{i=1}^na_ie_i^1\Big\|_*+\Big\|\sum_{i=1}^na_ie_i^2\Big\|_{**}$$ If $(e_n)_n$ admits a lower $\ell^1$-estimate of constant $c>0$ and $(e_n^2)_{n}$ does not admit any lower $\ell^1$-estimate then $(e_n^1)_{n}$ admits a lower $\ell^1$-estimate of the same constant $c$.

Suppose on the contrary that $(e_n^1)_{n}$ does not admit a lower $\ell^1$-estimate of constant $c$. Then there exist $\varepsilon
>0$, $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ with $\sum_{i=1}^n|a_i|=1$ such that $\|\sum_{i=1}^n a_i e_i^1\|_*<c-\varepsilon$. Also since $(e_n^2)_{n}$ does not admit any lower $\ell^1$-estimate, there exist $m\in{\mathbb{N}}$ and $b_1,\ldots,b_m\in{\mathbb{R}}$ such that $\sum_{j=1}^m|b_j|=1$ and $\| \sum_{j=1}^mb_j e_j^2
\|_{**}<\varepsilon/2$. Hence, we get that $$\label{gt2}\begin{split} \Big{\|}\sum_{i=1}^n\sum_{j=1}^m a_i\cdot
b_j e^1_{(i-1)m +j}\Big{\|}_*& \leq \sum_{j=1}^m |b_j|\Big{\|}
\sum_{i=1}^na_i e^1_{(i-1)m +j}\Big{\|}_* < c-\varepsilon
\end{split}$$ and similarly $$\label{gt3}\begin{split}
\Big{\|}\sum_{i=1}^n\sum_{j=1}^m a_i\cdot b_j e^2_{(i-1)m
+j}\Big{\|}_{**}& \leq \sum_{i=1}^n|a_i|\Big{\|}\sum_{j=1}^m b_j
e^2_{(i-1)m +j}\Big{\|}_{**} < \frac{\varepsilon}{2}
\end{split}$$ But then by (\[gt1\]), we obtain that $$\begin{split}\Big{\|}\sum_{i=1}^n\sum_{j=1}^m a_i\cdot b_j e_{(i-1)m +j}\Big{\|}_\circ
&\leq  \Big{\|}\sum_{i=1}^n\sum_{j=1}^m a_i\cdot b_j e^1_{(i-1)m +j}\Big{\|}_*\\
&+ \Big{\|}\sum_{i=1}^n\sum_{j=1}^m a_i\cdot b_j e^2_{(i-1)m
+j}\Big{\|}_{**} \stackrel{(\ref{gt2}),
(\ref{gt3})}{<}c-\frac{\varepsilon}{2}\end{split}$$ which since $\sum_{i=1}^n\sum_{j=1}^m |a_i|\cdot|b_j|=1$, contradicts that $(e_n)_n$ admits a lower $\ell^1$-estimate of constant $c$.

Let $k\in{\mathbb{N}}$ and $(x_s)_{s\in [{\mathbb{N}}]^k},(x_s^1)_{s\in
[{\mathbb{N}}]^k},(x_s^2)_{s\in [{\mathbb{N}}]^k}$ be three $k$-sequences in a Banach space $X$ such that for all $s\in[{\mathbb{N}}]^k$, $x_s=x_s^1+x_s^2$. Assume that the $k$-sequences $(x_s)_{s\in[{\mathbb{N}}]^k}$, $(x^1_s)_{s\in[{\mathbb{N}}]^k}$ and $(x^2_s)_{s\in[{\mathbb{N}}]^k}$ generate the sequences $(e_n)_{n}$, $(e_n^1)_{n}$ and $(e_n^2)_{n}$ respectively, as $k$-spreading models. If $(e_n)_n$ admits a lower $\ell^1$-estimate of constant $c>0$ and $(e_n^2)_{n}$ does not admit any lower $\ell^1$-estimate then $(e_n^1)_{n}$ admits a lower $\ell^1$-estimate of constant $c$.

For every $n\in{\mathbb{N}}$, $a_1,\ldots,a_n\in{\mathbb{R}}$ and $(s_j)_{j=1}^n$ in $[{\mathbb{N}}]^k$, we have $$\label{sd}\Big\|\sum_{j=1}^n
a_jx_{s_j}\Big\|\leq \Big\|\sum_{j=1}^n a_jx^1_{s_j}\Big\|+
\Big\|\sum_{j=1}^n a_jx^2_{s_j}\Big\|$$ Let $(E,\|\cdot\|_\circ),(E_1,\|\cdot\|_*),(E_2,\|\cdot\|_{**})$ be the seminormed linear spaces with Hamel bases $(e_n)_{n},(e_n^1)_{n}$ and $(e_n^2)_{n}$ respectively. Notice that (\[sd\]) implies that (\[gt1\]) holds and therefore the conclusion follows by Proposition \[propsplitl1\].

$k$-spreading models almost isometric to the $\ell^1$ basis
-----------------------------------------------------------

Let $c>0$, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence in a Banach space $X$. We will say that the $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ *generates* $\ell^1$ *as a $k$-spreading model of constant* $c$, if $(x_s)_{s\in[{\mathbb{N}}]^k}$ generates a $k$-spreading model $(e_n)_n$ which admits a lower $\ell_1$-estimate of constant $c$.

Let $X$ be a Banach space and $k\in{\mathbb{N}}$. Assume that $X$ admits a $k$-spreading model equivalent to the usual basis of $\ell^1$. Then for every $\varepsilon>0$ there exists a $k$-sequence $(y_s)_{s\in[{\mathbb{N}}]^k}$ in $X$ with $1-\varepsilon\leq \|y_s\|\leq 1$, for every $s\in
[{\mathbb{N}}]^k$, which generates $\ell^1$ as a $k$-spreading model of constant $1-\varepsilon$.

Let $(e_n)_n$ be a $k$-spreading model of $X$ which is equivalent to the usual basis of $\ell^1$. Also let $c=\inf
\|\sum_{j=1}^na_je_j\|$, taken over all $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ with $\sum_{j=1}^n|a_j|=1$. Let $\varepsilon>0$ and choose $0<\varepsilon'<c$, $p\in{\mathbb{N}}$ and $b_1,...,b_p$ in $[-1,1]$ with $\sum_{i=1}^p|b_i|=1$ such that $$\label{re}\frac{c-\varepsilon'}{c+2\varepsilon'}\geq 1-\varepsilon\;\;\text{and}\;\;\;
c\leq \Big\|\sum_{i=1}^p b_i e_i\Big\|\leq  c+\varepsilon'$$ Let $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a $k$-sequence is $X$ generating $(e_n)_n$ as a $k$-spreading model. By passing to an infinite subset $M$ of ${\mathbb{N}}$, we may assume that for every $n\in{\mathbb{N}}$, $a_1,\ldots,a_n\in[-1,1]$ and $(s_i)_{i=1}^n\in\textit{Plm}_n([M]^k)$ with $ s_1(1)\geq
M(n)$, we have $$\label{qk}\Bigg|\Big\|\sum_{i=1}^na_ix_{s_i}\Big\|-\Big\|\sum_{i=1}^na_i
e_i\Big\|\Bigg|\leq\varepsilon'\sum_{i=1}^n|a_i|$$ Hence by (\[re\]), for every $(s_i)_{i=1}^p
\in\textit{Plm}_p([M]^k)$ with $s_1(1)\geq M(p)$ we have that $$\label{lk}c-\varepsilon'\leq\Big\|\sum_{i=1}^p b_ix_{s_i}\Big\|\leq c+2\varepsilon'$$ For every $s=(n_1,...,n_k)\in[{\mathbb{N}}]^k$, we set $$\label{mk}y_s=\frac{\sum_{i=1}^pb_i x_{t_i^s}}{c+2{\varepsilon}'}
\;\text{where}\; t_i^s=\big(M(p\cdot n_j+i-1)\big)_{j=1}^k,\;\text{for all}\;1\leq i\leq p$$ Notice that $(t_i^{s})_{i=1}^p\in\textit{Plm}_p([{\mathbb{N}}]^k$ and $t_1^s(1)=M(p\cdot s(1))\geq M(p)$. Hence, by (\[re\]) and (\[lk\]), it is clear that $1-\varepsilon\leq \|y_s\|\leq 1$. Moreover, the $k$-subsequence $(y_s)_{s\in[{\mathbb{N}}]^k}$ generates $\ell^1$ as a $k$-spreading model of constant $1-\varepsilon$. Indeed, let $l\in{\mathbb{N}}$, $a_1,\ldots,a_l\in [-1,1]$ and $(s_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$ with $s_1(1)\geq l$. Notice that $(t_i^{s_1})_{i=1}^p \;^\frown\;...\;^\frown
{(t_i^{s_l})_{i=1}^p}\in\textit{Plm}_{p\cdot l}([{\mathbb{N}}]^k)$ and $t_1^{s_1}(1)=M(p\cdot s_1(1))\geq
    M(p\cdot l)$. Hence, $$\Big\|\sum_{j=1}^l a_j y_{s_j}\Big\|=
\Big\|\sum_{j=1}^la_j \cdot \sum_{i=1}^p \frac{ b_i
x_{t_i^{s_j}}}{c+2{\varepsilon}'}\Big\| \stackrel{(\ref{qk})}{\geq}
\frac{c-\varepsilon'}{c+2{\varepsilon}'}\sum_{j=1}^l\sum_{i=1}^p|a_j|\cdot|b_i|
\stackrel{(\ref{re})}{\geq}(1-\varepsilon)\sum_{j=1}^l|a_j|$$ and the proof is complete.

If we additionally assume that $X$ has a Schauder basis and $(x_s)_{s\in[M]^k}$ is plegma block (resp. plegma disjointly supported) then by (\[mk\]) it is easy to see that $(y_s)_{s\in[L]^k}$ is also plegma block (resp. plegma disjointly supported).

Plegma block generated $k$-spreading models equivalent to the $\ell^1$ basis
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It well known that if a Banach space $X$ with a Schauder basis admits an $\ell^1$ spreading model, then $X$ contains a block sequence which generates an $\ell^1$ spreading model. In this subsection we extend this result. More precisely, we have the following.

Let $X$ be a Banach space with a Schauder basis and $k\in{\mathbb{N}}$. Suppose that $\mathcal{SM}_k^{wrc}(X)$ contains up to equivalence the usual basis of $\ell^1$. Then there exists a plegma block generated $k$-spreading model of $X$ equivalent to the usual basis of $\ell^1$.

Let $k_{X}$ be the minimum of all $k\in{\mathbb{N}}$ such that the set $\mathcal{SM}_k^{wrc}(X)$ contains a sequence equivalent to the usual basis of $\ell^1$. By Remark \[remnj\], it suffices to show that $\mathcal{SM}_{k_{X}}(X)$ contains a sequence equivalent to the usual basis of $\ell_1$ which is plegma block generated. For $k_{X}=1$ this is a well known standard fact. So suppose that $k_{X}=k\geq 2$ and let $(e_n)_n\in\mathcal{SM}_{k}^{wrc}(X)$ be equivalent to the usual basis of $\ell^1$. By Corollary \[cor canonical tree with spr mod\], we may assume that $(e_n)_n$ is generated as a $k$-spreading model by a $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^{k}}$ which is subordinated with respect to the weak topology of $X$ and admits a canonical tree decomposition $(y_t)_{t\in[{\mathbb{N}}]^{\leq k}}$.

Let $(w_v)_{v\in [{\mathbb{N}}]^{k-1}}$ be the $(k-1)$-sequence in $X$ defined by $w_v=\sum_{t\sqsubseteq v}y_t$, for every $v\in[{\mathbb{N}}]^{k-1}$. Also let $(x'_s)_{s\in [{\mathbb{N}}]^k}$, be the $k$-sequence defined by $x'_s=w_{s|k-1}$, for every $s\in [{\mathbb{N}}]^k$. Notice that $(w_v)_{v\in [{\mathbb{N}}]^{k-1}}$ is subordinated with respect to the weak topology. Hence $(w_v)_{v\in [{\mathbb{N}}]^{k-1}}$ is a weakly relatively compact $(k-1)$-sequence. Also, by Lemma \[propxi\] we have that $(w_v)_{v\in [{\mathbb{N}}]^{k-1}}$ and $(x'_s)_{s\in [{\mathbb{N}}]^k}$ admit the same $(k-1)$-spreading models. Therefore, since the usual basis of $\ell_1$ is not contained up to equivalence in $\mathcal{SM}_{k-1}^{wrc}(X)$, we conclude that $(x'_s)_{s\in
[{\mathbb{N}}]^k}$ does not admit a $k$-spreading model equivalent to the usual basis of $\ell^1$. Since $x_s=x'_s+y_s$, for all $s\in [{\mathbb{N}}]^k$, by Corollary , we get that the $k$-sequence $(y_s)_{s\in[{\mathbb{N}}]^{k}}$ admits a $k$-spreading model equivalent to the usual basis of $\ell^1$. Since $(y_s)_{s\in[{\mathbb{N}}]^{k}}$ is a plegma block $k$-sequence in $X$ (see Proposition \[trocan\] (iii)), the proof is complete.

Duality of $c_0$ and $\ell^1$ $k$-spreading models
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It is well known that if a Banach space $X$ admits a $c_0$ spreading model, then $X^*$ admits an $\ell^1$ spreading model. In this subsection we extend this result.

\[lem for duality\] Let $X$ be a Banach space with a Schauder basis, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^{k}}$ be a $k$-sequence in $X$ which admits a canonical tree decomposition $(y_t)_{t\in[{\mathbb{N}}]^{\leq k}}$ and generates a $k$-spreading model equivalent to the usual basis of $c_0$. Then $y_\emptyset=0$ and there exist $1\leq j_0\leq k$ and $L\in[{\mathbb{N}}]^\infty$ such that the $k$-subsequence $(y_{s|j_0})_{s\in[L]^k}$ is plegma block and generates $c_0$ as a $k$-spreading model.

Since $(x_s)_{s\in[{\mathbb{N}}]^{k}}$ generates a $k$-spreading model, we have that $(x_s)_{s\in[{\mathbb{N}}]^{k}}$ is seminormalized. Let $(e_n)$ be the $k$-spreading model of $(x_s)_{s\in [{\mathbb{N}}]^k}$. Since $(e_n)_n$ is equivalent to the usual basis of $c_0$, we have that $(e_n)_n$ is Cesaro summable to zero. Using these observations we may easily conclude that $y_\emptyset=0$. We also observe that there exists $\delta>0$ such that for every $s\in[{\mathbb{N}}]^k$ there exists $1\leq
j\leq k$ such that $\|y_{s|j}\|>\delta$. Hence by Ramsey's theorem there exists $1\leq j_0\leq k$ and $L\in[{\mathbb{N}}]^\infty$ such that for every $s\in[L]^k$, $\|y_{s|j_0}\|>\delta$.

Let $n\in{\mathbb{N}}$, $a_1,\ldots,a_n\in{\mathbb{R}}$ and $(s_i)_{i=1}^n\in\textit{Plm}_n([L]^k)$. If $I$ is the interval of ${\mathbb{N}}$ with $\min I=\min\text{supp}(y_{s_1|j_0})$ and $\max I= \max
\text{supp}(x_{s_n|j_0})$, then Proposition \[trocan\] (v) and the fact that $y_\emptyset=0$, yield that $$I\Big(\sum_{i=1}^na_jx_{s_i}\Big)=\sum_{i=1}^na_iy_{s_i|j_0}$$ Hence if $C$ is the basis constant of the Schauder basis of $X$, we get that $$\frac{\delta}{2C}\max_{1\leq i\leq n}|a_i|\leq\Big\|\sum_{i=1}^na_iy_{s_i|j_0}\Big\|\leq 2C\Big\|\sum_{i=1}^na_ix_{s_i}\Big\|$$ Therefore, since $(x_s)_{s\in[L]^{k}}$ generates $c_0$ as a $k$-spreading model, we conclude that every $k$-spreading model of $(y_{s|j_0})_{s\in [L]^k}$ is equivalent to the usual basis of $c_0$.

The above lemma shows that the analogue of Theorem for the $c_0$ basis also holds. Namely we have the following.

\[getting block generated c\_0 spreading model\] Let $X$ be a Banach space with a Schauder basis and $k\in{\mathbb{N}}$. Suppose that $\mathcal{SM}_k^{wrc}(X)$ contains up to equivalence the usual basis of $c_0$. Then there exists a plegma block generated $k$-spreading model of $X$ equivalent to the usual basis of $c_0$.

Let $X$ be a Banach space. Assume that for some $k\in{\mathbb{N}}$ the set $\mathcal{SM}_k^{wrc}(X)$ contains a sequence equivalent to the usual basis of $c_0$. Then $X^*$ admits $\ell^1$ as a $k$-spreading model.

Let $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a subordinated $k$-sequence in $X$ generating $c_0$ as a spreading model. Let $Y$ separable subspace of $X$ containing the $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ and $T:Y\to
C[0,1]$ an isometry. Notice that $C[0,1]$ is a Banach space with a bimonotone Schauder basis and $(T(x_s))_{s\in[{\mathbb{N}}]^k}$ is subordinated. Let $({\varepsilon}_n)_n$ a null sequence of positive reals. By Theorem \[canonical tree\] there exist $L\in[{\mathbb{N}}]^\infty$ and a $k$-subsequence $(\widetilde{x}_s)_{s\in[L]^{k}}$ in $C[0,1]$ satisfying the following.

1.  $(\widetilde{x}_s)_{s\in[L]^k}$ admits a canonical tree decomposition $(\widetilde{y}_t)_{t\in[M]^{\leq k}}$.

2.  For every $s\in[L]^k$, $\|T(x_s)-\widetilde{x}_s\|<{\varepsilon}_n$, where $\min
      s=L(n)$.

Notice that property (P2) yields that $(\widetilde{x}_s)_{s\in[L]^{k}}$ generates $c_0$ as a $k$-spreading model. By Lemma \[lem for duality\] there exist $M\in[L]^\infty$ and $1\leq j_0\leq k$ such that the plegma block $k$-subsequence $(\widetilde{y}_{s|j_0})_{s\in[M]^k}$ generates $c_0$ as a $k$-spreading model. For every $s\in[M]^k$ we pick $\widetilde{y}_s^*\in S_{C[0,1]^*}$ with $\widetilde{y}_s^*(\widetilde{y}_{s|j_0})=\|\widetilde{y}_{s|j_0}\|$ and $\text{supp}\;\widetilde{y}_s^*\subseteq \text{range}\;
\widetilde{y}_{s|j_0}$. For every $s\in[M]^k$ we set $y^*_s=T^*(\widetilde{y}_s^*)$ and we choose $x^*_s$ in $X^*$ an extension of $y^*_s$ of the same norm. It is easy to check that $(x^*_s)_{s\in[M]^k}$ admits $\ell^1$ as a spreading model.

$k$-Cesàro summability vs $\ell^1$ $k$-spreading models
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In this section we extend the well known dichotomy of H.P. Rosenthal concerning Cesàro summability and $\ell^1$ spreading models (see also @AT, @M). We start by introducing the definition of the Cesàro summability for $k$-sequences in Banach spaces.

Definition of the $k$-Cesàro summability in Banach spaces
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\[Cesaro\] Let $X$ be a Banach space, $x_0\in X$ $k\in{\mathbb{N}}$, $(x_s)_{s\in
[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $M\in [{\mathbb{N}}]^\infty$. We will say that the $k$-subsequence $(x_s)_{s\in [M]^k}$ is $k$-Cesàro summable to $x_0$ if $$\Big(\substack{n\\ \\k}\Big)^{-1} \sum_{s\in [M|n]^k} x_s \;\;\substack{\|\cdot\| \\ \longrightarrow \\ n\to\infty}\;\; x_0$$ where $M|n=\{M(1),...,M(n)\}$.

\[rem on k-Cesaro summability\] Let $X$ be a Banach space, $x_0\in X$, $k\in{\mathbb{N}}$, $(x_s)_{s\in
[{\mathbb{N}}]^k}$ be a $k$-sequence in $X$ and $M\in [{\mathbb{N}}]^\infty$.

1.  If $(x_s)_{s\in[M]^k}$ norm converges to $x_0$, then $(x_s)_{s\in [M]^k}$ is $k$-Cesàro summable to $x_0$.

2.  If $(x_s)_{s\in[M]^k}$ is $k$-Cesàro summable to $x_0$ and in addition it is weakly convergent, then $x_0$ is the weak limit of $(x_s)_{s\in[M]^k}$.

3.  If $X^*$ is separable and for every $N\in[M]^\infty$, $(x_s)_{s\in[N]^k}$ is $k$-Cesàro summable to $x_0$ then there exists $L\in[M]^\infty$ such that $(x_s)_{s\in[L]^k}$ weakly converges to $x_0$.

Assertions (i) and (ii) are straightforward. For (iii), first observe that for every $x^*\in X^*$, ${\varepsilon}>0$ and $N\in[M]^\infty$ there exists an $L\in[N]^\infty$ such that $|x^*(x_s)-x^*(x_0)|<{\varepsilon}$, for all $s\in[L]^k$. Next for a norm dense subset $\{x^*_n:n\in{\mathbb{N}}\}$ of $X^*$, we inductively choose an $L\in[M]^\infty$ such that for every $n\in{\mathbb{N}}$ and $s\in[L]^k$ with $\min s\geq L(n)$ we have that $|x_i^*(x_s)-x_i^*(x_0)|<\frac{1}{n}$ for all $1\leq i=1 \leq n$. This yields that $(x_s)_{s\in[L]^k}$ weakly converges to $x_0$.

It is open if assertion (iii) of the above proposition remains valid without any restriction for $X^*$.

A density result for plegma families in $[{\mathbb{N}}]^k$
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In this subsection we will present a density Ramsey result concerning plegma families. For its proof, we will need the deep theorem of H. Furstenberg and Y. Katznelson @FK. Actually, we shall use the following finite version of this theorem (see also @G2).

\[Furstenberg and Katznelson Theorem\] Let $k\in{\mathbb{N}}$, $F$ be a finite subset of $\mathbb{Z}^k$ and $\delta>0$. Then there exists $n_0\in{\mathbb{N}}$ such that for all $n\geq n_0$, every subset $\mathcal{A}$ of $\{1,\ldots,n\}^k$ of size at least $\delta n^k $ has a subset of the form $a+d F$ for some $a\in \mathbb{Z}^k$ and $d\in{\mathbb{N}}$.

Our density result for plegma families is the following.

\[Lemma using Furstenberg and Katznelson Theorem to find long plegma\] Let $k, l\in{\mathbb{N}}$ and $\delta>0$. Then there exists $N_0\in {\mathbb{N}}$ such that for every $n\geq n_0$ and every subset $\mathcal{A}$ of $[\{1,\ldots,n\}]^k$ of size at least $\delta (\substack{n\\
k})$, there exists a plegma family $(s_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$ such that $s_j\in\mathcal{A}$, for every $1\leq j\leq l$.

For every $1\leq j\leq l$, let $t_j=\big(j,l+j,
2l+j,...,(k-1)l+j\big)$. Clearly $(t_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$. We set $F=\{\textbf{0}\}\cup\{t_j: \;1\leq j\leq l\}$, where $\textbf{0}=(0,...,0)$ is the zero element of $\mathbb{Z}^k$. Fix $\delta>0$. Since $\lim_{n}(\substack{n\\k})/n^k=1/k!$, there exists $m_0\in{\mathbb{N}}$ such that for every $n\geq m_0$ and every subset $\mathcal{A}$ of $[\{1,\ldots,n\}]^k$ of size at least $\delta (\substack{n\\ k})$ has density at least $\frac{\delta}{2k!}$ in $\{1,\ldots,n\}^k$. Hence, by Theorem \[Furstenberg and Katznelson Theorem\] (applied for $\frac{\delta}{2k!}$ in place of $\delta$) we have that there exists $n_0\geq m_0$ such that for every $n\geq n_0$, every subset $\mathcal{A}$ of $[\{1,\ldots,n\}]^k$ of size at least $\delta
(\substack{n\\k})$ has a subset of the form $a+d F$ for some $a\in
\mathbb{Z}^k$ and $d\in{\mathbb{N}}$. Notice that $a=a+d\textbf{0}\in\mathcal{A}$ and therefore $a\in
[\{1,...,n\}]^k$. For every $j\in\{1,...,l\}$, we set $s_j=a+dt_j$. Then $ \{s_j:\;1\leq j\leq l\}\subseteq
\mathcal{A}$. Moreover, since $a\in [{\mathbb{N}}]^k$ and $d\in{\mathbb{N}}$, we easily conclude that $(s_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$ and the proof is complete.

It is easy to see that for $k=1$ the preceding lemma trivially holds (it suffices to set $N_0=\lceil\frac{l}{\delta}\rceil$) and therefore Theorem \[Furstenberg and Katznelson Theorem\] is actually used for $k\geq 2$. However, it is not completely clear to us if the full strength of such a deep theorem like Furstenberg-Katznelson's is actually necessary for the proof of Proposition \[Lemma using Furstenberg and Katznelson Theorem to find long plegma\].

The main results
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\[Furstemberg's application on spreading models\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a bounded $k$-sequence in $X$. Let $M\in[{\mathbb{N}}]^\infty$ such that the subsequence $(x_s)_{s\in[M]^k}$ generates a Cesàro summable to zero $k$-spreading model $(e_n)_n$. Then for every $L\in[M]^\infty$ the $k$-subsequence $(x_s)_{s\in[L]^k}$ is $k$-Cesàro summable to zero.

Assume on the contrary that there exists $L\in[M]^\infty$ such that $(x_s)_{s\in[L]^k}$ is not $k$-Cesàro summable to zero. Then there exists a $\theta>0$ and a strictly increasing sequence $(p_n)_n$ of natural numbers such that for every $n\in{\mathbb{N}}$, $$\label{wq}\Big(\substack{p_n\\ \\ k}  \Big)^{-1}\Big{\|}\sum_{s\in
[L|p_n]^k} x_s\Big{\|}>\theta$$ For each $n\in{\mathbb{N}}$, we pick $x_n^*\in S_{X^*}$ such that $x_n^*\big( (\substack{p_n\\ \\ k})^{-1} \sum_{s\in
[L|p_n]^k} x_s  \big)>\theta$ and we set $$\label{yt}\mathcal{A}_n=\Big\{s\in
\big[\{1,...,p_n\}\big]^k:\;x_n^*(x_{L(s)})>\frac{\theta}{2}\Big\}$$ where $S_{X^*}$ is the unit sphere of $X^*$. By (\[wq\]) and a simple averaging argument we easily derive that $|\mathcal{A}_n|\geq
\frac{\theta}{2K} \Big(\substack{p_n\\ \\ k}  \Big)$, where $K=\sup\{\|x_s\|:s\in [{\mathbb{N}}]^k\}$.

We fix $m\in{\mathbb{N}}$. By Proposition \[Lemma using Furstenberg and Katznelson Theorem to find long plegma\], with $\delta=\frac{\theta}{2K}$ and $l=2m-1$, there exists $n_0\in{\mathbb{N}}$ such that for every $n\geq n_0$ there exists a plegma family $(s_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$ such that $\{s_j:\;1\leq j\leq l\}\subseteq \mathcal{A}_{n}$. Therefore setting $t_i=L(s_{m+i-1})$ for all $1\leq i\leq m$, we conclude that for every $m\in{\mathbb{N}}$ there exists $(t_i)_{i=1}^m\in\textit{Plm}_m([L]^k)$ such that $t_1(1)\geq L(m)$ and $\Big\|\frac{1}{m}\sum_{j=1}^m
x_{t_j}\Big\|>\frac{\theta}{2}$. This easily yields that $(e_n)_n$ is not Cesàro summable to zero, which is a contradiction.

\[Rosenthal prop\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a bounded $k$-sequence in $X$. Let $M\in[{\mathbb{N}}]^\infty$ such that the subsequence $(x_s)_{s\in[M]^k}$ generates an unconditional $k$-spreading model $(e_n)_n$. Then at least one of the following holds:

1.  The sequence $(e_n)_n$ is equivalent to the usual basis of $\ell^1$.

2.  For every $L\in [M]^\infty$ $(x_s)_{s\in [L]^k}$ is $k$-Cesàro summable to zero.

Assume that $(e_n)_n$ is not equivalent to the usual basis of $\ell^1$. Since $(e_n)_n$ is an unconditional spreading sequence, by Proposition \[equiv forms for 1-subsymmetric weakly null\] we have that $(e_n)_n$ is Cesàro summable to zero. Hence, by Proposition \[Furstemberg's application on spreading models\] we have that $(x_s)_{s\in [L]^k}$ is $k$-Cesàro summable to zero, for every $L\in [M]^\infty$.

Notice that in the case $k=1$ the two alternatives of Corollary \[Rosenthal prop\] are mutually exclusive. This does not remain valid for $k\geq 2$. For instance, assume that in Example \[example\], $(e_n)_n$ is the usual basis of $\ell^1$. Then the basis $(x_s)_{s\in[{\mathbb{N}}]^{k+1}}$ of $X$ generates a $(k+1)$-spreading model equivalent to the usual basis of $\ell^1$ and simultaneously for every $L\in[{\mathbb{N}}]^\infty$, $(x_s)_{s\in[L]^{k+1}}$ is $(k+1)$-Cesàro summable to zero. Indeed, let $L\in[{\mathbb{N}}]^\infty$ and $n\in{\mathbb{N}}$. Then since every plegma tuple in $[L|n]^{k+1}$ is of size less than $n$, we have $$\Big\|\Big(\substack{n\\\;\\k+1}\Big)^{-1}\sum_{s\in [L|n]^{k+1}} x_s\Big\|_{k+1}\leq n\Big(\substack{n\\\;\\k+1}\Big)^{-1}$$ Since $k+1\geq 2$, $\lim_n n(\substack{n\\k+1})^{-1}=0$. Thus for every $L\in[{\mathbb{N}}]^\infty$, $(x_s)_{s\in [L]^{k+1}}$ is Cesàro summable to zero.

\[Rosenthal thm\] Let $X$ be a Banach space, $k\in{\mathbb{N}}$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ be a weakly relatively compact $k$-sequence in $X$. Then there exists $M\in [{\mathbb{N}}]^\infty$ such that at least one of the following holds:

1.  The subsequence $(x_s)_{s\in[M]^k}$ generates a $k$-spreading model equivalent to the usual basis of $\ell^1$.

2.  There exists $x_0\in X$ such that for every $L\in
    [M]^\infty$, $(x_s)_{s\in [L]^k}$ is $k$-Cesàro summable to $x_0$.

First we notice that if there exists $M\in [{\mathbb{N}}]^\infty$ such that $(x_s)_{s\in[M]^k}$ norm converges to some $x_0\in X$, then by Proposition \[rem on k-Cesaro summability\] (i), we immediately get that (2) holds. So we may suppose for the sequel that the $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ does not contain any norm convergent $k$-subsequence.

Let $M_1\in[{\mathbb{N}}]^\infty$ such that $(x_s)_{s\in[M_1]^k}$ generates a $k$-spreading model $(e_n)_n$. By Proposition \[Create subordinated\] there exists $M_2\in[M_1]^\infty$ such that $(x_s)_{s\in[M_2]^k}$ is subordinated (with respect to the weak topology). Let $\widehat{\varphi}:[M_2]^k\to (X,w)$ be the continuous map witnessing this and $x_0=\widehat{\varphi}(\emptyset)$.

For every $s\in [M_2]^k$ we set $x'_s=x_s-x_0$. Notice that the map $\widehat{\psi}:[M_2]^k\to (X,w)$ defined by $\widehat{\psi}(t)=\widehat{\varphi}(t)-x_0$ is continuous. Hence $(x'_s)_{s\in[M_2]^k}$ is subordinated. Since $\widehat{\psi}(\emptyset)=0$, by Proposition \[subordinating yields convergence\], we have that $(x'_s)_{s\in[M_2]^k}$ is weakly null. Moreover, since $(x_s)_{s\in[{\mathbb{N}}]^k}$ does not contain any norm convergent $k$-subsequence, it is easy to see that $(x'_s)_{s\in[M_2]^k}$ is seminormalized.

Let $(e'_n)_{n}$ be a $k$-spreading model of $(x'_s)_{s\in
[M_2]^k}$ and let $M\in [M_2]^\infty$ such that $(x'_s)_{s\in[M]^k}$ generates $(e'_n)_{n}$. By Theorem \[unconditional spreading model\], $(e'_n)_{n}$ is unconditional and therefore, by Corollary \[Rosenthal prop\], we have that either $(e'_n)_{n}$ is equivalent to the usual basis of $\ell^1$ or for every $L\in [M]^\infty$, $(x'_s)_{s\in [L]^k}$ is $k$-Cesàro summable to zero. Since $x_s=x'_s+x_0$, for every $s\in [M]^k$, by Lemma \[triv-ell\] we have that the first alternative yields that $(e_n)_n$ is equivalent to the usual basis of $\ell^1$ while the second one, easily gives that for every $L\in [M]^\infty$, $(x_s)_{s\in [L]^k}$ is $k$-Cesàro summable to $x_0$.

The $k$-spreading models of $c_0$ and $\ell^p$, $1\leq p<\infty$ {#spreading models of c_0 and l^p}
================================================================

In this section we deal with a natural problem, posed to us by Th. Schlumprecht, of determining the spreading models of the classical sequence spaces. As we will see, while the spreading models of $\ell^p$, $1\leq
p<\infty$, are as expected, the class of the 2-spreading models of $c_0$ is surprising large.

The $k$-spreading models of $c_0$
---------------------------------

It is well known that every non trivial spreading model of $c_0$ generates a space isomorphic to $c_0$. On the other hand the class of the 2-spreading models of $c_0$ is quite large. As we will see $\mathcal{SM}_2(c_0)$ contains all bimonotone Schauder basic spreading sequences. Notice that this property of $c_0$ is similar to the one of $C(\omega^\omega)$ admitting every $1$-unconditional spreading sequence as a spreading model (see @Odell).

We start with the following lemma.

\[on spr mod of c\_0 second lem\] Let $(e_n)_n$ be a spreading sequence in $\ell^\infty$ and let $(x_s)_{s\in [{\mathbb{N}}]^2}$ be the $2$-sequence in $c_0$ defined by $x_s=(e_{s(1)}(1), e_{s(1)}(2),...,e_{s(1)}(s(2)), 0,0,...)$, for every $s\in [{\mathbb{N}}]^2$. Then for every non trivial $2$-spreading model $(\widetilde{e}_n)_{n}$ of $(x_s)_{s\in[{\mathbb{N}}]^2}$, $l\in{\mathbb{N}}$ and $a_1,\ldots,a_l\in{\mathbb{R}}$, we have $$\label{gto}\Big\|\sum_{i=1}^la_ie_i\Big\|_\infty\leq\Big\|\sum_{i=1}^la_i\widetilde{e}_i\Big\|\leq\max_{1\leq
j\leq l}\Big\|\sum_{i=j}^l a_i e_i\Big\|_\infty$$

We fix $l\in{\mathbb{N}}$ and $a_1,\ldots, a_l\in{\mathbb{R}}$. It is easy to check that for every $(s_i)_{i=1}^l\in
\textit{Plm}_l([{\mathbb{N}}]^k)$, we have that $$\label{kh}\Big\|\sum_{i=1}^la_ix_{s_i}\Big\|_\infty\leq\max_{1\leq j\leq
l}\Big\|\sum_{i=j}^l a_i e_{s_i(1)} \Big\|_\infty$$ Let $M\in[{\mathbb{N}}]^\infty$ such that $(x_s)_{s\in[M]^2}$ generates a non trivial $2$-spreading model $(\widetilde{e}_n)_{n}$. Then by (\[kh\]), we easily obtain the righthand inequality of (\[gto\]). To complete the proof, we fix ${\varepsilon}>0$ and $m_{\varepsilon}\in{\mathbb{N}}$ such that $$\label{pi}\Big\|\sum_{i=1}^la_ie_i\Big\|_\infty-{\varepsilon}\leq
\Big|\sum_{i=1}^la_ie_i(m_{\varepsilon})\Big|$$ Notice that for every $(s_i)_{i=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^2)$ in $[{\mathbb{N}}]^2$ with $
s_1(1)\geq m$, we have that $$\label{pii}\Big|\sum_{i=1}^la_ie_i(m)\Big|\leq\Big\|\sum_{i=1}^la_ix_{s_i}\Big\|_\infty$$ Therefore, since $(x_s)_{s\in[M]^2}$ generates $(\widetilde{e}_n)_{n}$ as a 2-spreading model, by (\[pi\]) and (\[pii\]), we get that $$\Big\|\sum_{i=1}^la_ie_i\Big\|_\infty-{\varepsilon}\leq \Big|\sum_{i=1}^la_ie_i(m_{\varepsilon})\Big|\leq \Big\|\sum_{i=1}^la_i\widetilde{e}_i\Big\|_\infty$$ Since this holds for every ${\varepsilon}>0$, we obtain the lefthand inequality of (\[gto\]) and the proof is complete.

\[c\_0 has every bimonotone as order 2\] For every Schauder basic spreading sequence $(e_n)_n$ there exists $(\widetilde{e}_n)_{n}\in\mathcal{SM}_2(c_0)$ equivalent to $(e_n)_n$. In particular, if $(e_n)_n$ is bimonotone then $(e_n)_{n}$ is contained in $\mathcal{SM}_2(c_0)$.

We may assume that $(e_n)_n$ is a sequence in $\ell^\infty$. Let $C>0$ be the basis constant of $(e_n)_n$. By Lemma \[on spr mod of c\_0 second lem\] there exists $(\widetilde{e}_n)_{n}\in\mathcal{SM}_2(c_0)$ such that for all $l\in{\mathbb{N}}$ and $a_1,\ldots,a_l\in{\mathbb{R}}$, we have $$\label{lo}\Big\|\sum_{i=1}^la_ie_i\Big\|_\infty\leq\Big\|\sum_{i=1}^la_i\widetilde{e}_i\Big\|\leq\max_{1\leq
j\leq l}\Big\|\sum_{i=j}^l a_i e_i\Big\|_\infty\leq
(1+C)\Big\|\sum_{i=1}^la_ie_i\Big\|_\infty$$ Hence, $(e_n)_n$ and $(\widetilde{e}_n)_{n}$ are equivalent. Moreover, if in addition $(e_n)_n$ is bimonotone then $\text{max}_{1\leq j\leq
l}\|\sum_{i=j}^l a_i e_i\|_\infty\leq \|\sum_{i=1}^l a_i
e_i\|_\infty$ and therefore $(\widetilde{e}_n)_{n}$ is isometric to $(e_n)_n$.

\[c\_0 universcal for singular\] For every singular spreading sequence $(e_n)_n$, there exists $(\widetilde{e}_n)_n\in\mathcal{SM}_2(c_0)$ equivalent to $(e_n)_n$.

Let $e_n=e'_n+e$ be the natural decomposition of $(e_n)_n$. By Remark \[properties of the natural decomposition\], $(e'_n)_n$ is spreading and 1-unconditional. Hence, by Proposition \[c\_0 has every bimonotone as order 2\], there exists a 2-sequence $(x_s)_{s\in[{\mathbb{N}}]^2}$ in $c_0$ generating $(e'_n)_n$ as a 2-spreading model. For every $s\in[{\mathbb{N}}]^2$, let $\widetilde{x}_s$ be the sequence in $c_0$ defined by $\widetilde{x}_s(1)=\|e\|$ and $\widetilde{x}_s(n+1)=x_s(n)$ for all $n\in{\mathbb{N}}$. It is easy to see that $(\widetilde{x}_s)_{s\in[{\mathbb{N}}]^2}$ generates a 2-spreading model $(\widetilde{e}_n)_n$, satisfying $$\Big\|\sum_{j=1}^na_j\widetilde{e}_j\Big\|=\max\Big\{\Big|\sum_{j=1}^na_j\Big|\cdot\|e\|,\Big\|\sum_{j=1}^na_je'_j\Big\|\Big\}$$ for all $n\in{\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$. Therefore, by Remark \[properties of the natural decomposition\], we conclude that $(e_n)_n$ and $(\widetilde{e}_n)_{n}$ are equivalent.

By Proposition \[c\_0 has every bimonotone as order 2\] and Corollary \[c\_0 universcal for singular\] we have the following.

\[universality\_of\_c\_0\] The set $\mathcal{SM}_2(c_0)$ is isomorphically universal for all spreading sequences.

The $k$-spreading models of $\ell^p$, for $1\leq
p<\infty$
------------------------------------------------

The $k$-spreading models of the spaces $\ell^p$, for $1\leq p<\infty$, can be treated as the classical spreading models. This is based on the observation that the usual basis of these spaces is symmetric. Therefore, the norm-behavior of the $k$-sequences admitting a canonical tree decomposition is identical with the one of sequences being of the form $(x_n+x)_n$, where $(x_n)_n$ is block.

Especially, for the case of $\ell^1$, one has to make use of the $w^*$-relative compactness of the bounded $k$-sequences in order to pass to a subordinated $k$-subsequence with respect to the $w^*$-topology and in turn to a further one which is approximated by a $k$-subsequence admitting a canonical tree decomposition. This procedure yields the following.

Let $1\leq p<\infty$ and $(\widetilde{e}_n)_n$ be a $k$-spreading model of $\ell^p$, for some $k\in{\mathbb{N}}$. Then there exist $a_1,a_2\geq 0$ such that $(\widetilde{e}_n)_n$ is isometric to the sequence $(a_1e_1+a_2e_{n+1})_n$, where $(e_n)_n$ denotes the usual basis of $\ell^p$. More precisely we have the following.

1.  The sequence $(\widetilde{e}_n)_n$ is trivial if and only if $a_2=0$.

2.  The sequence $(\widetilde{e}_n)_n$ is singular if and only if $a_1\neq0$ and $a_2\neq0$.

3.  The sequence $(\widetilde{e}_n)_n$ is Schauder basic if and only if $a_1=0$ and $a_2\neq0$. In this case $(\widetilde{e}_n)_n$ is equivalent to the usual basis of $\ell^p$.

It can also be shown that every $(\widetilde{e}_n)_n\in\mathcal{SM}_{k}^{wrc}(c_0)$, satisfies the analogue of Theorem with $c_0$ in place of $\ell^p$.

Every non trivial $k$-spreading model of $\ell^p$, $1<p<\infty$, generates a space isometric to $\ell^p$. In particular, every non trivial $k$-spreading model of $\ell^1$ is Schauder basic and equivalent to the usual basis of $\ell^1$.

A reflexive space not admitting $\ell^p$ or $c_0$ as a spreading model {#space Odel_Schlum}
======================================================================

A space not admitting any $\ell^p$, for $1\leq p<\infty$, or $c_0$ spreading model was constructed in @O-S. In the same paper it is asked if there exists a space which does not contain any $\ell^p$, for $1\leq p<\infty$, or $c_0$ $k$-iterated spreading model of any $k\in{\mathbb{N}}$. In this section we give an example of a reflexive space $X$ answering affirmatively this problem.

The definition of the space $X$
-------------------------------

The construction of $X$ is closely related to the corresponding one in @O-S. Let $(n_j)_j$ and $(m_j)_j$ be two strictly increasing sequences of natural numbers satisfying the following:

1.  $\sum_{j=1}^\infty \frac{1}{m_j}\leq 0,1$.

2.  For every $a>0$, we have that $\frac{n_j^a}{m_j}\stackrel{j\to\infty}{\longrightarrow}\infty$.

3.  For every $j\in{\mathbb{N}}$, we have that $\frac{n_j}{n_{j+1}}<\frac{1}{m_{j}}$.

Let $\|\cdot\|$ be the norm on $c_{00}({\mathbb{N}})$, implicitly defined as follows. For every $x\in c_{00}({\mathbb{N}})$ we set $$\label{eq15}
  \|x\|=\max \Big\{ \|x\|_\infty,\big( \sum_{j=1}^\infty \|x\|_j^2
\big)^\frac{1}{2}\Big\}$$ where $\|x\|_j=sup\{\frac{1}{m_j}\sum_{q=1}^{n_j}\|E_q(x)\|:E_1<\ldots<E_{n_j}\}$.

Let $X$ be the completion of $c_{00}({\mathbb{N}})$ under the above norm. It is easy to see that the Hamel basis of $c_{00}({\mathbb{N}})$ is an unconditional basis of the space $X$. Also notice that for every $x\in X$ the sequence $w=(\|x\|_j)_j$ belongs to $\ell^2$ and $\big( \sum_{j=1}^\infty \|x\|_j^2
\big)^\frac{1}{2}=\|w\|_{\ell^2}\leq \|x\|$.

The main results
----------------

The following is the main result of this section.

\[Odel slumpr theorem\] For every $k\in{\mathbb{N}}$ and $(e_n)_n\in\mathcal{SM}_k(X)$, the space $E$ generated by $(e_n)_n$ does not contain any isomorphic copy of $\ell^p$, $1\leq
  p<\infty$, or $c_0$.

Given the above theorem we get the following consequence which the aforementioned problem stated in @O-S.

For every $k\in{\mathbb{N}}$, the spaces generated by the $k$-iterated spreading models of $X$ do not contain any isomorphic copy of $\ell^p$, $1\leq
p<\infty$, or $c_0$.

By Theorem \[Odel slumpr theorem\] and James' Theorem we have that for every $k\in{\mathbb{N}}$, the spaces generated by the unconditional $k$-spreading models of $X$ are reflexive. By Corollary \[qwqwe\] we have that for every $k\in{\mathbb{N}}$, every space generated by a $k$-iterated spreading model of $X$ is isomorphic to the space generated by an unconditional $k$-spreading model of $X$. By Theorem \[Odel slumpr theorem\], the proof is complete.

Also notice this example shows that Krivine's theorem @Kr concerning $\ell^p$ or $c_0$ block finite representability cannot be captured by the notion of $k$-spreading models.

Proof of Theorem \[Odel slumpr theorem\]
----------------------------------------

We will need the next well known lemma (see @AT).

\[small\_estimation\_on\_means\] Let $j<j_0$ in ${\mathbb{N}}$ and $(x_q)_{q=1}^{n_{j_0}}$ be a block sequence in the unit ball $B_X$ of $X$. Then $$\Big\| \frac{x_1+\ldots+x_{n_{j_0}}}{n_{j_0}}
  \Big\|_j<\frac{2}{m_j}$$

\[small means\] Let $d_0<j_0$ in ${\mathbb{N}}$, and $(x_q)_{q=1}^{n_{j_0}}$ be a block sequence in $B_X$. We set $E=\{n\in{\mathbb{N}}:n>d_0\}$ and $w_q=(\|x_q\|_j)_j$, for all $1\leq q\leq n_{j_0}$. Assume that for some $0<{\varepsilon}<1$ there exists a disjointly supported finite sequence $(w'_q)_{q=1}^{n_{j_0}}$ in $\ell^2$ such that $\|E(w_q-w'_q)\|_{\ell^2}<{\varepsilon}$, for all $1\leq q\leq n_{j_0}$. Then $$\Big\|\frac{x_1+\ldots+x_{n_{j_0}}}{n_{j_0}}\Big\|<0.2+{\varepsilon}+2n_{j_0}^{-\frac12}$$

By Lemma \[small\_estimation\_on\_means\], we have that $$\Bigg\|\Big(\Big\|\frac{\sum_{q=1}^{n_{j_0}}x_q}{n_{j_0}}\Big\|_j \Big)_{j=1}^{d_0}\Bigg\|_{\ell^2}
  \leq\sum_{j=1}^{d_0}\Big\|\frac{\sum_{q=1}^{n_{j_0}}x_q}{n_{j_0}}\Big\|_j\leq\sum_{j=1}^{d_0}\frac{2}{m_j}<0,2$$ Using the above and the observation that $\|E(w'_q)\|_{\ell^2}\leq2$, for all $1\leq q\leq n_{j_0}$, we get the following. $$\begin{split}
    \Bigg\|\Big(\Big\|\frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}x_q \Big\|_j\Big)_j\Bigg\|_{\ell_2}&
    \leq0,2+\Bigg\|\Big(\Big\|\frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}x_q\Big\|_j\Big)_{j>d_0}\Bigg\|_{\ell^2}\\
    &\leq0,2+\Bigg\|\frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}\big(w_q(j)\big)_{j>d_0}\Bigg\|_{\ell^2}
    \leq0,2+\Big\|\sum_{q=1}^{n_{j_0}}\frac{E(w'_q)}{n_{j_0}}\Big\|_{\ell^2}+{\varepsilon}\\
    &\leq0,2+\Big(\sum_{q=1}^{n_{j_0}}\Big(\frac{2}{n_{j_0}}\Big)^2\Big)^\frac{1}{2}+{\varepsilon}= 0,2+{\varepsilon}+2n_{j_0}^{-\frac12}
  \end{split}$$ Moreover $\|\frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}x_q\|_\infty\leq\frac{1}{n_{j_0}}<\frac{1}{m_1}<0,1$. Hence by (\[eq15\]) the proof is completed.

For all $k\in{\mathbb{N}}$, every plegma block generated $k$-spreading model of $X$ is not equivalent to the usual basis of $\ell^1$.

Assume on the contrary that there exist $k\in{\mathbb{N}}$ and a plegma block $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ in $X$ which generates $\ell^1$ as a $k$-spreading model. By Proposition , we may also assume that $x_s\in B_X$, for all $s\in[{\mathbb{N}}]^k$ and $(x_s)_{s\in[{\mathbb{N}}]^k}$ generates $\ell^1$ as a $k$-spreading model of constant $1-{\varepsilon}$, where ${\varepsilon}=0,1$.

For every $s\in[{\mathbb{N}}]^k$, let $w_s=(\|x_s\|_j)_j$. Since $(w_s)_{s\in[{\mathbb{N}}]^k}$ is a $k$-sequence in $B_{\ell^2}$, it is weakly relatively compact. Hence, by Proposition \[cor for subordinating\], there exists $M\in[{\mathbb{N}}]^\infty$ such that the $k$-subsequence $(w_s)_{s\in[M]^k}$ is subordinated with respect the weak topology on $\ell^2$. Let $\widehat{\varphi}:[M]^{\leq
k}\to(\ell^2,w)$ be the continuous map witnessing this. By Theorem \[canonical tree\], there exist $L\in[M]^\infty$ and a $k$-subsequence $(\widetilde{w}_s)_{s\in[L]^k}$ in $X$ satisfying the following.

1.  $(\widetilde{w}_s)_{s\in[L]^k}$ admits a canonical tree decomposition $(\widetilde{z}_t)_{t\in[L]^{\leq k}}$ with $\widetilde{z}_\emptyset=\widehat{\varphi}(\emptyset)$.

2.  For every $s\in[L]^k$, $\|w_s-\widetilde{w}_s\|_{\ell^2}<{\varepsilon}/2$, where $\min s=L(n)$.

3.  The $k$-subsequence $(\widetilde{w}_s)_{s\in[L]^k}$ is subordinated with respect to the weak topology of $\ell^2$.

Let $d_0\in{\mathbb{N}}$ such that $\|E(\widehat{\varphi}(\emptyset))\|_{\ell^2}<\frac{{\varepsilon}}{2}$, where $E=\{d_0+1,\ldots\}$. For every $s\in[L]^k$ we set $w'_s=\widetilde{w}_s-\widehat{\varphi}(\emptyset)$. By Proposition \[trocan\] (iv), we have that $(w'_s)_{s\in[L]^k}$ is plegma disjointly supported. Moreover, notice that $\|E(w_s-w'_s)\|_{\ell^2}<{\varepsilon}$, for all $s\in[L]^k$. We pick $j_0>d_0$ such that $2n_{j_0}^{-\frac12}<{\varepsilon}$. Since $(x_s)_{s\in[{\mathbb{N}}]^k}$ generates $\ell^1$ as a $k$-spreading model of constant $0,9$, we may choose $(s_q)_{q=1}^{n_{j_0}}\in\textit{Plm}_{n_{j_0}}([L]^k)$ such that $$\label{eq16}
\Big\| \frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}x_{s_q} \Big\|\geq 0,8$$ Observe that $d_0,j_0,{\varepsilon}$, $(x_{s_q})_{q=1}^{n_{j_0}}$ and $(w'_{s_q})_{q=1}^{n_{j_0}}$ satisfy the assumptions of Lemma \[small means\]. Hence $$\Big\|
\frac{1}{n_{j_0}}\sum_{q=1}^{n_{j_0}}x_{s_q}
\Big\|<0,2+{\varepsilon}+2n_{j_0}^{-\frac12}<0,4$$ which contradicts (\[eq16\]) and the proof is complete.

The space $X$ is reflexive.

Lemma implies that the space $X$ does not contain any isomorphic copy of $\ell^1$. Moreover, using that $\frac{n_j}{m_j}\stackrel{j\to\infty}{\longrightarrow}\infty$, it is easy to see that the space $X$ does not contain any isomorphic copy of $c_0$. Since the basis of $X$ is unconditional, the result follows by James' theorem.

\[corell1\] For all $k\in{\mathbb{N}}$, every $k$-spreading model of $X$ is not equivalent to the usual basis of $\ell^1$.

Suppose on the contrary that there exist $k\in{\mathbb{N}}$ and a bounded $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ which generates a $k$-spreading model equivalent to the $\ell^1$ basis. By the reflexivity of $X$, we have that $(x_s)_{s\in[{\mathbb{N}}]^k}$ is weakly relatively compact. Therefore, by Theorem , there exists a plegma block generated $k$-spreading model of $X$ equivalent to the usual basis of $\ell^1$, which contradicts to Lemma .

Let $1<p\leq\infty$. Then for every $\delta,C>0$ there exists $l_0\in{\mathbb{N}}$ such that for every $l\geq l_0$ and every block sequence $(x_q)_{q=1}^{n_l}$ in $X$ with $\|x_q\|>\delta$, for all $1\leq q\leq
  n_l$, we have that $$\Big\|\sum_{q=1}^{n_l}x_q\Big\|>Cn_l^\frac{1}{p}$$ where by convection $\frac1\infty=0$

Since $\frac{n_l^{1-\frac{1}{p}}}{m_l}\stackrel{l\to\infty}{\longrightarrow}\infty$, there exists $l_0\in{\mathbb{N}}$ such that $\frac{n_l^{1-\frac{1}{p}}}{m_l}>\frac{C}{\delta}$, for every $l\geq l_0$. Let $(x_q)_{q=1}^{n_l}$ be a block sequence in $X$ with $\|x_q\|>\delta$, for all $1\leq q\leq
  n_l$. Then $$\Big\|\sum_{q=1}^{n_l}x_q\Big\|\geq\Big\|\sum_{q=1}^{n_l}x_q\Big\|_l\geq
  \frac{1}{m_l}\sum_{q=1}^{n_l}\|x_q\|>\frac{n_l}{m_l}\delta>Cn_l^\frac{1}{p}$$

\[adfs\] For all $k\in{\mathbb{N}}$, every $k$-spreading model of $X$ is not equivalent to the usual basis of $\ell^p$, $1<p<\infty$, or $c_0$.

Suppose on the contrary that for some $k\in{\mathbb{N}}$, $X$ admits a $k$-spreading model $(e_n)_n$, which is equivalent to the usual basis of either $\ell^p$, for some $1<p<\infty$, or $c_0$. First we shall treat the case of $\ell^p$. Since $X$ is reflexive, we have that $(e_n)_n\in\mathcal{SM}_k^{wrc}(X)$. By Corollary \[cor canonical tree with spr mod\], there exists a subordinated $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ admitting a canonical tree decomposition $(y_t)_{t\in[{\mathbb{N}}]^{\leq k}}$, which generates $(e_n)_n$ as a $k$-spreading model. Since the basis of $X$ is unconditional and $(e_n)_n$ is Cesáro summable to zero, it is easy to see that $y_\emptyset=0$. Notice that $(x_s)_{s\in[{\mathbb{N}}]^k}$ is seminormalized and let $\delta>0$ such that $\|x_s\|>\delta$, for all $s\in[{\mathbb{N}}]^k$. Hence, for every $s\in[{\mathbb{N}}]^k$ there exists $1\leq d\leq k$ such that $\|y_{s|d}\|>\frac{\delta}{k}$. By Ramsey's theorem there exists $1\leq d\leq
  k$ and $L\in[{\mathbb{N}}]^\infty$ such that for every $s\in[L]^k$, $\|y_{s|d}\|>\frac{\delta}{k}$. By Proposition \[trocan\] (iii), we have that $(y_{s|d})_{s\in[L]^k}$ is plegma block. Fix $C>0$. By Lemma we have that there exists $l_0$ such that for every $l>l_0$ and $(s_q)_{q=1}^{n_l}\in\textit{Plm}_{n_l}[L]^k$ we have that $\Big\|\sum_{q=1}^{n_l}y_{s_q|d}\Big\|>Cn_l^{\frac1p}$. Hence, dy the 1-unconditionality of the basis of $X$, we conclude that $$\Big\|\sum_{q=1}^{n_l}x_{s_q}\Big\|>Cn_l^{\frac1p}$$ Since the above holds for every $C>0$ we have that $(e_n)_n$ is not equivalent to the usual basis of $\ell^p$, which is a contradiction.

Finally, if $(e_n)_n$ is equivalent to the usual basis of $c_0$, then the proof is carried out using identical arguments as above and applying Lemma for $p=\infty$.

\[Proof of Theorem \[Odel slumpr theorem\]\] Suppose that for some $k\in{\mathbb{N}}$ there exists $(e_n)_n\in\mathcal{SM}_{k}(X)$ such that the space $E$ generated by $(e_n)_n$ contains an isomorphic copy of $Y$, where $Y$ is either $\ell^p$, for some $1\leq p<\infty$, or $c_0$. Obviously $(e_n)_n$ is non trivial. Since $X$ is reflexive, $(e_n)_n\in\mathcal{SM}_{k}^{wrc}(X)$. By Corollary , we have that $\mathcal{SM}_{k+1}(X)$ contains a sequence equivalent to the usual basis of $Y$. By Corollaries \[corell1\] and \[adfs\], we get the contradiction.

A space $X$ such that $\mathcal{SM}_k(X)$ is a proper subset of $\mathcal{SM}_{k+1}(X)$ {#s12}
=======================================================================================

In this section we shall present a Banach space $\mathfrak{X}_{k+1}$, having an unconditional basis $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ which generates a $(k+1)$-spreading model equivalent to the usual basis of $\ell^1$, while the space $\mathfrak{X}_{k+1}$ does not admit $\ell^1$ as a $k$-spreading model. Moreover, $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ is not $(k+1)$-Cesàro summable to any $x_0$ in $\mathfrak{X}_{k+1}$.

The definition of the space $\mathfrak{X}_{k+1}$
------------------------------------------------

We fix for the following a positive integer $k$. We will need the next definition.

A family $\mathcal{P}\subseteq [{\mathbb{N}}]^{k+1}$ will be called plegmatic in $[{\mathbb{N}}]^{k+1}$, if there exist a finite block sequence $F_1<\ldots<F_{k+1}$ of subsets of ${\mathbb{N}}$ with $|F_1|=\ldots=|F_{k+1}|$ such that $\mathcal{P}\subseteq
F_1\times\ldots\times F_{k+1}$. A plegmatic family $\mathcal{P}\subseteq[{\mathbb{N}}]^{k+1}$ will be called Schreier if in addition $|F_1|\leq \min F_1$.

For instance, for every $(s_j)_{j=1}^l\in\textit{Plm}_l({\mathbb{N}}]^{k+1}$, the family $\mathcal{P}=\{s_1,\ldots,s_l\}$ is plegmatic but notice that not all plegmatic families in $[{\mathbb{N}}]^{k+1}$ are plegma.

Let $(e_s)_{s\in [{\mathbb{N}}]^{k+1}}$ be the Hamel basis of $c_{00}([{\mathbb{N}}]^{k+1})$. For every $x=\sum_{s\in[{\mathbb{N}}]^{k+1}}x(s)e_s$ in $c_{00}([{\mathbb{N}}]^{k+1})$, we set $$\label{no}\|x\|=\sup
\Big(\sum_{i=1}^n\|\mathcal{P}_i(x)
\|_1^2\Big)^{\frac{1}{2}}$$ where $\|\mathcal{P}(x)\|_1=\sum_{s\in \mathcal{P}}|x(s)|$, for all $\mathcal{P}\subseteq [{\mathbb{N}}]^{k+1}$ and the supremum in (\[no\]) is taken over all finite sequences $(\mathcal{P}_i)_{i=1}^n$ of disjoint Schreier plegmatic families in $[{\mathbb{N}}]^{k+1}$. The space $\mathfrak{X}_{k+1}$ is defined to be the completion of $(c_{00}([{\mathbb{N}}]^{k+1}),\|\cdot\|)$.

The proof of the next proposition is straightforward.

\[mn\]The Hamel basis $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ of $c_{00}([{\mathbb{N}}]^{k+1})$ is an unconditional basis for the space $\mathfrak{X}_{k+1}$ and it generates a $(k+1)$-spreading model which is isometric to the usual basis of $\ell^1$.

We may also define a norming set $W$ for the space $\mathfrak{X}_{k+1}$ as follows. First, let $$W^0=\Big\{
\sum_{s\in\mathcal{P}} \pm e_s^*: \mathcal{P}\subseteq
[{\mathbb{N}}]^{k+1}\;\;\text{is Schreier plegmatic}\Big\}$$ For each $f=\sum_{s\in\mathcal{P}} e_s^*\in W^0$, the support of $f$, denoted by $\text{supp}(f)$, is defined to be the family $\mathcal{P}$. It is easy to see that a norming set for $\mathfrak{X}_{k+1}$ is the set $W$ which consists of all $f=\sum_{i=1}^n\lambda_i f_i$ where $(f_i)_{i=1}^n$ is a sequence in $W^0$ such that $\text{supp}(f_i)\cap\text{supp}(f_j)=\emptyset$, for all $1\leq
i<j\leq n$ and $\sum_{i=1}^n\lambda_i^2\leq 1$.

In order to study the basic properties of the space $\mathfrak{X}_{k+1}$, we need the following proposition.

Every plegma disjointly generated $k$-spreading model of $\mathfrak{X}_{k+1}$ is not equivalent to the usual basis of $\ell^1$.

The proof is postponed in the next subsection. Assuming Proposition we are able to prove the following.

The space $\mathfrak{X}_{k+1}$ has the next properties.

It is reflexive.

There is no sequence $(e_n)_n\in\mathcal{SM}_{k}(\mathfrak{X}_{k+1})$ equivalent to the usual basis of $\ell^1$.

Every (k+1)-subsequence of $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ is not $(k+1)$-Cesàro summable to any $x_0$ in $\mathfrak{X}_{k+1}$.

\(i) By Proposition \[mn\], we have that $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ is unconditional. Also, it is easy to check that it is boundedly complete. Thus $c_0$ is not contained in $\mathfrak{X}_{k+1}$. Moreover, the same holds for $\ell^1$, since otherwise there would exist a disjointly supported sequence $(x_n)_{n}\in\mathfrak{X}_{k+1}$ equivalent to the usual basis of $\ell^1$, which is impossible by Proposition . Hence, by James' theorem @J, the space $\mathfrak{X}_{k+1}$ is reflexive.\
(ii) Assume on the contrary, that there exists $(e_n)_{n}$ in $\mathcal{SM}_k(\mathfrak{X}_{k+1})$ equivalent to the usual basis of $\ell^1$. Since $\mathfrak{X}_{k+1}$ is reflexive, we get that $(e_n)_{n}\in\mathcal{SM}^{wrc}_k(\mathfrak{X}_{k+1})$. Hence, by Corollary \[cor canonical tree with spr mod\], $(e_n)_{n}$ is generated by a $k$-sequence $(x_s)_{s\in[{\mathbb{N}}]^k}$ in $X_{k+1}$ admitting a canonical tree decomposition $(y_t)_{t\in[{\mathbb{N}}]^{\leq
k}}$. Setting $x'_s=x_s-y_\emptyset$, for all $s\in[{\mathbb{N}}]^k$, by Lemma \[triv-ell\], we have that $(x'_s)_{s\in[{\mathbb{N}}]^k}$ also admits a $k$-spreading model equivalent to the usual basis of $\ell^1$. Since $(x'_s)_{s\in[{\mathbb{N}}]^k}$ is a plegma disjointly supported $k$-sequence, by Proposition we have reached to a contradiction.\
(iii) Since $\mathfrak{X}_{k+1}$ is reflexive we have that $(e_s)_{s\in[{\mathbb{N}}]^{k+1}}$ is a weakly null $(k+1)$-sequence. Let $M\in[{\mathbb{N}}]^\infty$ and assume that $(e_s)_{s\in[M]^{k+1}}$ is $(k+1)$-Cesàro summable to some $x_0\in\mathfrak{X}_{k+1}$. By Proposition \[rem on k-Cesaro summability\](ii), we get that $x_0=0$. For every $n\in{\mathbb{N}}$, let $$y_n=\Big(
\substack{(k+2)n\\\ k+1} \Big)^{-1}\sum_{s\in[M|
(k+2)n]^{k+1}}e_s$$ where $l_n=(k+2)n$, $\mathcal{P}_n=F_1^n\times\ldots\times
  F_{k+1}^n$, where for every $1\leq i\leq
k+1$, $F_i^n=\{M(in+1),\ldots,M((i+1)n)\}$ and $f_n=\sum_{s\in
\mathcal{P}_n}e^*_s$. It is easy to check that $$\label{qr}f_n(y_n)= n^{k+1}\cdot \Big( \substack{(k+2)n\\k+1}
\Big)^{-1}\stackrel{n\to\infty}{\longrightarrow}\frac{(k+1)!}{(k+2)^{k+1}}$$ Since $\|y_n\|\geq f_n(y_n)$, by (\[qr\]) we conclude that $(e_s)_{s\in[M]^{k+1}}$ is not $(k+1)$-Cesàro summable to $x_0=0$, a contradiction.

Proof of Proposition 
---------------------

\[supports\] Let $x\in \mathfrak{X}_{k+1}$ of finite support and $f\in W^0$ such that $\text{supp} (f)\cap
\text{supp}(x)\neq \emptyset$. Then $|\text{supp}(f)|\leq
n_0^{k+1}$, where $n_0=\max\{s(1):\;s\in\text{supp}(x)\}$.

There exist $F_1<\ldots< F_{k+1}$ subsets of ${\mathbb{N}}$ such that $|F_1|=...=|F_{k+1}|$, $\text{supp}(f)\subseteq
F_1\times\ldots\times F_{k+1}$ and $|F_1|\leq\min F_1$. Hence $|\text{supp}(f)|\leq(\min F_1)^{k+1}$. Let $s\in\text{supp}(f)\cap\text{supp}(x)$. Then $n_0\geq s(1)\geq\min
F_1$. Hence $n_0\geq\min F_1$ and therefore $|\text{supp}(f)|\leq
n_0^{k+1}$.

Let $N_0\in{\mathbb{N}}$. Then for every $0<{\varepsilon}<1$, every $l\in{\mathbb{N}}$ and every disjointly supported finite sequence $(x_j)_{j=1}^l$ in the unit ball of $\mathfrak{X}_{k+1}$ such that for every $1\leq j\leq l$ and $s\in\text{supp}(x_j)$, $s(1)\leq N_0$, we have that $$\Big\|\frac{1}{l}\sum_{j=1}^{l}x_{j}\Big\|\leq {\varepsilon}+\frac{N_0^{k+1}}{{\varepsilon}^2 l}$$

We fix $0<{\varepsilon}<1$, $l\in{\mathbb{N}}$ and $(x_j)_{j=1}^l$ satisfying the assumptions of the lemma. Let $\varphi=\sum_{i=1}^n\lambda_if_i\in W$, where $n\in{\mathbb{N}}$, $\lambda_1,\ldots,\lambda_n\in{\mathbb{R}}$ with $\sum_{i=1}^n\lambda_i^2\leq1$ and $f_1,\ldots,f_n\in W^0$ pairwise disjointly supported. For every $j=1,\ldots,l$ we set $$I_j=\Big\{i\in\{1,\ldots,n\}:\;\text{supp}(f_i)\cap\text{supp}(x_j)\neq\emptyset \Big\}$$ By Lemma \[supports\], we have that for every $1\leq j\leq l$, if $i\in I_j$ then $|\text{supp}(f_i)|\leq N_0^{k+1}$. Also let $F_1=\{j\in\{1,\ldots,l\}:\; \sum_{i\in I_j}\lambda_i^2
<\varepsilon^2\}$ and $F_2=\{1,\ldots,l\}\setminus F_1$. It is easy to see that $\sum_{i\in I_j}\frac{f_i(x_j)}{( \sum_{i\in I_j}f_i(x_j)^2 )^\frac{1}{2}}f_i$ belongs to $W$, for all $1\leq j\leq l$. Hence, since $\|x_j\|\leq1$, we have that $\sum_{i\in I_j} f_i(x_j)^2\leq1$, for all $1\leq j\leq l$. Therefore we have $$\begin{split}
\varphi\Big(\sum_{j=1}^{l}x_j\Big)&= \sum_{i=1}^n\lambda_i f_i\Big(\sum_{j=1}^{l}x_j\Big) =
    \sum_{j=1}^{l} \sum_{i=1}^n \lambda_i f_i(x_j)\\
    &= \sum_{j=1}^{l} \sum_{i\in I_j} \lambda_i f_i(x_j)\leq
    \sum_{j=1}^{l}
    \Big{(}\sum_{i\in I_j} \lambda_i^2\Big{)}^\frac{1}{2} \Big{(}\sum_{i\in I_j}f_i(x_j)^2\Big{)}^\frac{1}{2}\\
    &\leq  \sum_{j\in F_1} \Big{(}\sum_{i\in I_j} \lambda_i^2\Big{)}^\frac{1}{2}  + \sum_{j\in F_2}
     \Big{(}\sum_{i\in I_j} \lambda_i^2\Big{)}^\frac{1}{2}\\
    &\leq \varepsilon |F_1|+|F_2|\leq {\varepsilon}l+|F_2|
  \end{split}$$ If for some $1\leq i\leq n$ we have that $J_i\neq\emptyset$ then, by Lemma \[supports\] we have that $|\text{supp}(f_i)|\leq N_0^{k+1}$ and since $(x_j)_{j=1}^{l}$ are disjointly supported, we conclude that $|J_i|\leq N_0^{k+1}$. Therefore, for every $1\leq i\leq n$, $|J_i|\leq N_0^{k+1}$. Hence $${\varepsilon}^2 |F_2|\leq \sum_{j\in F_2}\sum_{i\in I_j}\lambda_i^2\leq \sum_{j=1}^{l}\sum_{i\in I_j}\lambda_i^2=\sum_{i=1}^n|J_i|\lambda_i^2\leq N_0^{k+1}\sum_{i=1}^n \lambda_i^2\leq
N_0^{k+1}$$ which yields that $|F_2|\leq N_0^{k+1}/{\varepsilon}^2$. Therefore, for every $\varphi\in W$ we have $$\varphi\Big(\sum_{j=1}^{l}x_j\Big)\leq \varepsilon
l+\frac{N_0^{k+1}}{{\varepsilon}^2}$$ Since $W$ is a norming set for $\mathfrak{X}_{k+1}$, the proof is complete.

\(i) Let $\mathcal{G}_1,\mathcal{G}_2\subseteq
[{\mathbb{N}}]^{k+1}$. We will call the pair $(\mathcal{G}_1,\mathcal{G}_2)$ weakly plegmatic if for every $s_2\in G_{2}$ there exists $s_1\in
\mathcal{G}_i$ such that the pair $\{s_1,s_2\}$ is plegmatic.\
(ii) For every $0\leq j\leq l$, let $\mathcal{G}_j\subseteq [{\mathbb{N}}]^{k+1}$. The finite sequence $(\mathcal{G}_j)_{j=0}^l$ will be called a *weakly plegmatic path of subsets of* $[{\mathbb{N}}]^{k+1}$, if for every $0\leq i< l$ the pair $(\mathcal{G}_{i},\mathcal{G}_{i+1})$ is weakly plegmatic.

\[blocking the first coordinates in weakly allowable paths\] Let $(\mathcal{G}_j)_{j=0}^k$ be a weakly plegmatic path of subsets in $[{\mathbb{N}}]^{k+1}$. Then $\max\{s(1):s\in\cup_{j=0}^k
\mathcal{G}_j\}\leq \max\{ s(k+1): s\in {\mathcal{G}}_0\}$.

Let $0\leq j\leq k$ and $s\in {\mathcal{G}}_j$. Then it is easy to see that there exists a sequence $(s_i)_{i=0}^j$ in $[{\mathbb{N}}]^{k+1}$ with $s_i\in {\mathcal{G}}_i$, for every $0\leq
i\leq j-1$ and $s_j=s$, such that $\{s_i,s_{i+1}\}$ is plegmatic, for all $0\leq i\leq j-1$. Hence $$s(1)=s_j(1)<s_{j-1}(2)<...<s_0(j+1)\leq s_0(k+1)\leq \max\{
s(k+1): s\in {\mathcal{G}}_0\}$$

\[Lemma for 2 vectos of norm almost 2\] Let $0<\eta<\frac18$ and $x_1,x_2\in\mathfrak{X}_{k+1}$ with disjoint finite supports such that $\|x_1\|,\|x_2\|\leq1$ and $\|x_1+x_2\|> 2-2\eta$. Let $\mathcal{G}_1\subseteq\text{supp}(x_1)$ such that $\|\mathcal{G}_1^c(x_1)\|\leq \eta$. Then there exists $\mathcal{G}_2\subseteq\text{supp}(x_2)$ satisfying the following.

1.  The pair $(\mathcal{G}_1,\mathcal{G}_2)$ is a weakly plegmatic path and

2.  $\|\mathcal{G}_2^c(x_2)\|\leq \eta^\frac{1}{8}$.

Since $\|x_1+x_2\|>2-2\eta$, there exists $\varphi\in W$ such that $\varphi(x_1+x_2)>2-2\eta$. Since $\|x_1\|,\|x_2\|\leq1$, we get that $\varphi(x_1)>1-2\eta$ and $\varphi(x_2)>1-2\eta$. The functional $\varphi$ is of the form $\sum_{i=1}^n \lambda_i f_i$, where $f_1,\ldots,f_n$ are pairwise disjoint supported elements of $W^0$ and $\sum_{i=1}^n\lambda_i^2\leq1$. We set $I=\{1,\ldots,n\}$ and we split it to $I_1$ and $I_2$ as follows: $$I_1=\{i\in I:\;\text{supp}(f_i)\cap {\mathcal{G}}_1\neq\emptyset\}\;\;\text{and}\;\;I_2=I\setminus I_1=\{i\in I:\;\text{supp}(f_i)\subseteq {\mathcal{G}}_1^c\}$$ We also set $\varphi_1=\sum_{i\in I_1}\lambda_i f_i$ and $\varphi_2=\sum_{i\in I_2}\lambda_i f_i$. Hence $\varphi_2(x_1)\leq\|{\mathcal{G}}_1^c(x_1)\|\leq \eta$ and therefore $\varphi_1(x_1)> 1-3\eta$. Applying Cauchy-Schwartz's inequality we get that $$1-3\eta < \varphi_1(x_1)=\sum_{i\in I_1}\lambda_i f_i(x_1)\leq
  \Big{(} \sum_{i\in I_1}\lambda_i^2 \Big{)}^\frac{1}{2}\Big{(} \sum_{i\in I_1}f_i(x_1)^2
   \Big{)}^\frac{1}{2}\leq\Big{(} \sum_{i\in I_1}\lambda_i^2 \Big{)}^\frac{1}{2}$$ Since $\sum_{i\in I}\lambda_i^2\leq1$, we have that $(\sum_{i\in I_2}\lambda_i^2)^\frac{1}{2}< (1-(1-3\eta)^2)^\frac{1}{2}\leq (6\eta)^\frac{1}{2}$. Hence $$\varphi_2(x_2)=\sum_{i\in I_2}\lambda_if_i(x_2)\leq \Big{(}
     \sum_{i\in I_2}\lambda_i^2 \Big{)}^\frac{1}{2}\Big{(} \sum_{i\in I_2} f_i(x_2)^2
      \Big{)}^\frac{1}{2}< (6\eta)^\frac{1}{2}$$ Hence $\varphi_1(x_2)> 1-2\eta-(6\eta)^\frac{1}{2}>1-4\eta^\frac{1}{2}$. We set ${\mathcal{G}}_2=\text{supp}(x_2)\cap\text{supp}(\varphi_1)$. Then by the definition of $I_1$ it is immediate that the pair $({\mathcal{G}}_1,{\mathcal{G}}_2)$ is weakly plegmatic. Finally, since $\|{\mathcal{G}}_2(x_2)\|^2+\|{\mathcal{G}}_2^c(x_2)\|^2\leq
\|x_2\|^2\leq 1$ and $\|{\mathcal{G}}_2(x_2)\|\geq
\varphi_1(x_2)$, we get that $\|{\mathcal{G}}_2^c(x_2)\|\leq (1-(1-4\eta^\frac12)^2)^\frac{1}{2}<\eta^\frac18$ and the proof is complete.

An iterated use of the above yields the following.

\[final lemma for mathfrak-X-\_k+1\] Let $m\in{\mathbb{N}}$ and $0<{\varepsilon}<\frac18$ Then for every sequence $(x_i)_{i=0}^m$ of disjointly and finitely supported vectors in $\mathfrak{X}_{k+1}$ with $\|x_i\|\leq1$, for all $0\leq i\leq m$, and $\|x_i+x_{i+1}\|> 2-2\varepsilon^{8^m}$, for all $0\leq i<m$, there exists a weakly plegmatic path $({\mathcal{G}}_i)_{i=0}^m$ of subsets of $[{\mathbb{N}}]^{k+1}$ such that $\mathcal{G}_i\subseteq \text{supp}\;x_i$ and $\|{\mathcal{G}}_i^c(x_i)\|<{\varepsilon}$, for all $0\leq i\leq m$.

We are now ready to give the proof of Proposition .

\[Proof of Proposition :\] Assume on the contrary that the space $\mathfrak{X}_{k+1}$ admits a plegma disjointly generated $k$-spreading model equivalent to the usual basis of $\ell^1$. Let $0<{\varepsilon}<\frac18$. By Proposition and Remark there exists a sequence $(x_t)_{t\in[{\mathbb{N}}]^k}$ in the unit ball of $\mathfrak{X}_{k+1}$ which is plegma disjointly supported and generates $\ell^1$ as a $k$-spreading model of constant $c>1-{\varepsilon}^{8^k}$. Therefore, we may suppose that $$\label{eq14}
    \Big\|\frac1l\sum_{j=1}^lx_{t_j}\Big\|>1-{\varepsilon}^{8^k}$$ for all $l\in{\mathbb{N}}$ and $(t_j)_{j=1}^l\in\textit{Plm}_l([{\mathbb{N}}]^k)$ with $t_1(1)\geq l$.

We set $t_0=\{2,4,\ldots,2k\}$, $N_0=\max\{s(k+1):s\in\text{supp}(x_{t_0})\}$ and $L=\{2n:s>k\}$. For every $t\in[L]^k$ we select ${\mathcal{G}}_t\subseteq[{\mathbb{N}}]^k$ such that ${\mathcal{G}}_t\subseteq\text{supp}(x_t)$, $\|{\mathcal{G}}_t^c(x_t)\|<{\varepsilon}$ and $s(1)<N_0$, for all $s\in{\mathcal{G}}_t$, as follows. Let $t\in[L]^k$. Observe $t\in [{\mathbb{N}}]_\shortparallel^{k}$ and $t_0<t$. By Proposition \[accessing everything with plegma path of length |s\_0|\] there exists a plegma path $(t_j)_{j=0}^{k}$ in $[{\mathbb{N}}]^{k}$, with $t_k=t$. By Corollary \[final lemma for mathfrak-X-\_k+1\] (for $m=k$) there exists a weakly plegmatic path $({\mathcal{G}}_j)_{j=0}^{k}$ such that ${\mathcal{G}}_j\subseteq \text{supp}\;x_{t_j}$ and $\|{\mathcal{G}}_j^c(x_{t_j})\|<{\varepsilon}$, for all $j=0,\ldots,k$. We set ${\mathcal{G}}_t={\mathcal{G}}_k$. Lemma \[blocking the first coordinates in weakly allowable paths\] and Corollary \[final lemma for mathfrak-X-\_k+1\] yield that the choice of $({\mathcal{G}}_t)_{t\in[L]^k}$ is as desired.

For every $t\in[L]^k$, let $x_t^1={\mathcal{G}}_t(x_t)$. Then $\|x_t-x_t^1\|<{\varepsilon}$, for all $t\in[L]^k$. Hence by (\[eq14\]) we get that for every $l\in{\mathbb{N}}$ and every $(t_j)_{j=1}^l\in\textit{Plm}_l([L]^k)$ with $t_1(1)\geq l$, we have that $$\label{eq9}
    \Big{\|} \frac{1}{l}\sum_{i=1}^l x^1_{t_i} \Big{\|}>1-2\varepsilon>\frac68$$ Moreover notice that $(x'_t)_{t\in[L]^k}$ is a plegma disjointly supported $k$-subsequence in the unit ball of $\mathfrak{X}_{k+1}$. Therefore, by Lemma \[blocking the first coordinates in weakly allowable paths\] and (\[eq9\]) for $l>8N_0^{k+1}/5{\varepsilon}^2$, we get a contradiction. The proof of Proposition is complete.

As we have mentioned in the introduction of this article, the $k$-spreading models of a Banach space $X$ have a transfinite extension yielding an hierarchy of $\xi$-spreading models, for $\xi<\omega_1$. It can be shown that the space in Section \[space Odel\_Schlum\] does not admit $\ell^p$, for $1\leq p<\infty$, or $c_0$ as $\xi$-spreading model, for every $\xi<\omega_1$. Also an analogue of the last example exists. Namely, for every limit countable ordinal $\xi$ there exists a reflexive space $X_\xi$ admitting $\ell^1$ as $\xi$-spreading model but not less.

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[^1]: 2010 *Mathematics Subject Classification*: 46B03, 46B06, 46B25, 46B45, 05D10

[^2]: *Keywords*: Spreading models, Ramsey theory

[^3]: This research is partially supported by NTUA Programme PEBE 2009 and it is part of the PhD Thesis of the third named author

[^4]: A sequence $(e_n)_{n}$ in a seminormed space $(E,\|\cdot\|_*)$ is called spreading if for every $n\in{\mathbb{N}}$, $k_1<\ldots<k_n$ in ${\mathbb{N}}$ and $a_1,\ldots,a_n\in{\mathbb{R}}$ we have that $\|\sum_{j=1}^na_j e_j\|_*=\|\sum_{j=1}^n a_j
    e_{k_j}\|_*$.

    In the literature the term "spreading model" usually indicates the space generated by the corresponding spreading sequence rather than the sequence itself. We have chosen to use the term for the spreading sequence and whenever we refer to $\ell^p$ or $c_0$ spreading model we shall mean that the spreading sequence is equivalent to the usual basis of the corresponding space.

{'source': 'arXiv_out/1105'}