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数理解析研究所講究録
第2081巻 2018年 62-68

A COMMENT ON BAGARIA‐SHELAH’S FRAGMENT OFMARTIN’S AXIOMTERUYUKI YORIOKA

INTRODUCTION

Harrington‐Shelah showed that, for x \in \mathbb{R} , if \aleph_{1^{L[x\rfloor}=\aleph_{1} and Martin’s axiom

MA holds, then there exists a \triangle_{3}^{1}(x) ‐set of reals without Baire property [4]. Bagaria\aleph_{1} , MA( $\sigma$‐centered) holds
extended their result as follow: For x \in \mathbb{R} , if \aleph_{1^{L[x]}and every Aronszajn tree is special, then there exists a \triangle_{3}^{1}(x) ‐set of reals without
Baire property [1]. Harrington‐Shelah also showed that, if \aleph_{1} is inaccessible to the=reals and MA holds, then \aleph_{1} is weakly compact in L , and that the existence ofa weakly compact cardinal is equiconsistent to the assertion that MA holds andevery projective set of reals has the Baire property [4]. So in [1], Bagaria asked
if MA( $\sigma$‐centered) plus the assertion that every Aronszajn tree is special implies
MA( $\sigma$ ‐linked). In [2], Bagaria‐Shelah proved the consistency of the assertion (^{*} )that MA( $\sigma$ ‐centered) holds, every Aronszajn tree is special, and MA ( $\sigma$ ‐linked) fails.
In this paper, the consistency of the assertion (^{*} ) is proved by use of the idea
due to Bagaria Shelah in [2] combining with the rectangle refining property due
to Larson‐Todorčevič [6]. §1 provides Bagaria‐Shelah’s work in [2] and introduces$\Gamma$their fragment of Martin’s axiom. §2 provides some remarks of the rectangle refining
property and a proof of the theorem in this paper. In §3, some previous works on
fragments of Martin’s axiom are mentioned and are compared to the theorem inthis paper.
1. BAGARIA−SHELAH’S FRAGMENT OF MARTIN’S AXIOM

In [2], Bagaria‐Shelah introduced the following property of forcing notions.
Definition 1.1 ([2, DEFINITION 1

For an integer k \geq2,

a forcing notion \mathb {P}satisfies the property \mathrm{P}\mathrm{r}_{k} if, for any \{p_{ $\alpha$} : $\alpha$\in$\omega$_{1}\} \in [\mathb {P}]^{\aleph_{1} , there exists a pairwise
disjoint uncountable family \{u $\xi$ : $\xi$\in $\omega$\} of non‐empty finite subsets of $\omega$_{1} such that,for each, there exists \langle$\alpha$_{i} : i\in k\rangle
such that\{$\xi$_{i} : i\in k\}\in[$\omega$_{1}]^{k}has a common extension in \mathbb{P}.

\displayst le\in prod_{i\nk}u_{$\xi$_{\mathrm{z}\{p_{$\alpha$_{i} : i\in k\}

MA (\mathrm{P}\mathrm{r}_{k}) denotes the forcing axiom for forcing notions with the property \mathrm{P}\mathrm{r}_{k}.

The property \mathrm{P}\mathrm{r}_{k} is stronger than the countable chain condition. A $\sigma$‐centeredforcing satisfies the property \mathrm{P}\mathrm{r}_{k} for every integer k \geq 2 . Bagaria‐Shelah provedthat a specialization of an Aronszajn tree by finite approximations also satisfies theproperty \mathrm{P}\mathrm{r}_{k} for every integer k \geq 2 [ 2 , LEMMA 2]. So, for every integer k \geq 2,
MA (\mathrm{P}\mathrm{r}_{k}) implies MA ( $\sigma$ ‐centered) and the assertion that every Aronszajn tree is
special. They also showed that, for any integer k\geq 2 , the property \mathrm{P}\mathrm{r}_{k} is preserved

under finite support iterations [2, LEMMA 3].

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In [2], Bagaria‐Shelah introduced the following forcing notion that plays a role ofthe failure of MA( $\sigma$‐linked) in the extension with finite support iterations of forcing
notions with the property \mathrm{P}\mathrm{r}_{k}.

Definition 1.2 ([2, LEMMA 4

For an integer k\geq 2 , the forcing notion \mathb {P}_{*}^{k} con‐sists of triples p= \{u_{p}, A_{p}, h_{p}\} such that\bullet\bullet\bulletu_{p} is a finite subset of $\omega$_{1},A_{p} is a subset of [u_{p}]^{k+1} (which is a finite set),h_{p} is a function from the set \{v\subseteq u_{p}: [v]^{k+1}\cap A_{p}=\emptyset\} into $\omega$ such that,for every l\in \mathrm{r}\mathrm{a}\mathrm{n}(h_{p}) and $\rho$\in [h_{p}^{-1}[\{l\}]]^{k}, \cup $\rho$ belongs to dom(hp),ordered by: q\leq_{\mathbb{P}_{*}^{k} p iff u_{q}\supseteq u_{p}, A_{p}=A_{q}\cap[u_{p}]^{k+1} and h_{q}\supseteq h_{p}.
Note that \mathb {P}_{*}^{k} is of size \aleph_{1} . Bagaria‐Shelah proved that \mathb {P}_{*}^{k} has precaliber \aleph_{1} [2,LEMMA 4 (1)]. They also define the following \mathb {P}_{*}^{k} ‐names.

Definition 1.3 ([2, LEMMA 4
the \mathb {P}_{*}^{k} ‐name \dot{A}_{k}^{*} byLet k be an integer not smaller than 2. Define\dot{A}_{*}^{k}:=\{(\check{v},p):p\in \mathbb{P}_{*}^{k}, v\in A_{p}\},and define the \mathb {P}_{*}^{k} ‐name \dot{\mathb {Q} _{*}^{k} by\dot{\mathbb{Q} _{*}^{k} :=\{ (\check{v}, p) : p\in \mathbb{P}_{*}^{k}, v\in \mathrm{d}\mathrm{o}\mathrm{m}(h_{p})\}.
We notice that1\displayst le\vdash_{\mathb {P}_{*^k}(\dot{\mathcal{A}_{*}^k}=\bigcup_{ \inG_{\mathrm{F}_{*^k} A_{p} \dot{\mathb {Q} _{*}^{k}=\{v\in[$\omega$_{1}]^{<\aleph_{0} : [v]^{k+1}\cap\dot{\mathcal{A} _{*}^{k}=\emptyset\}and`,

By considering \dot{\mathb {Q} _{*}^{k} as a \mathb {P}_{*}^{k} ‐name for a forcing notion, ordered \mathrm{b}\mathrm{y}\supseteq , Bagaria‐Shelah
proved that

[2, LEMMA 4 (3)]: |\vdash_{\mathb {P}_{\mathrm{r} ^{k} (\dot{\mathb {Q} _{*}^{k} is $\sigma$-k ‐linked

and[2, LEMMA 4 (5)]: 1\vdash_{\mathb {P}_{*}^{k}for any \{v_{ $\alpha$} : $\alpha$\in$\omega$_{1}\} \in [\dot{\mathb {Q} _{*}^{k}]^{\aleph_{1} with v_{ $\alpha$} \not\in $\alpha$ , andany pairwise disjoint uncountable family \{u_{ $\xi$} : $\xi$\in $\omega$\} of non‐empty finite
subsets of $\omega$_{1} , there exists \{$\xi$_{i}:i\in k+1\} \in [$\omega$_{1}]^{k+1} such that, for every
does not belong to \dot{\mathb {Q} _{*}^{k} ”\langle$\alpha$_{i}:i\in k+1\rangle(\displaystyle\in\prod_{i\nk+1}u_{$\xi$_{2} , \displayst le\bigcup_{i\nk+1}v_{$\alpha$_{i}The last assertion implies that|\vdash_{\mathb {P}_{*}^{k} \dot{\mathb {Q}_{*}^{k} is not $\sigma$-(k+1) ‐linked”
Definition 1.4 ([2, LEMMA 4\mathb {P}_{*}^{k} ‐name \dot{I}_{$\alpha$} such that

For an integer k \geq

2

and $\alpha$ \in $\omega$_{1} , define the

1\vdash_{\mathrm{P}_{*}^{k} \dot{I}_{ $\alpha$}:=\{v\in\dot{\mathb {Q} _{*}^{k}:v\not\subset $\alpha$\}Note that

1\vdash_{\mathb {P}_{*}^{k} ` \`{I}_{$\alpha$} is dense in \dot{\mathb {Q}_{*}^{k} ”[2, LEMMA 4 (4)]. The following is a key point of the proof of the failure ofMA( $\sigma$ ‐linked) in the extension with finite support iterations of forcing notions withthe property \mathrm{P}\mathrm{r}_{k}.

64Lemma 1.5 ([2, LEMMA 6

For any integer k \geq

2

and any \mathb {P}_{*}^{k} ‐name \dot{\mathb {Q} for a

forcing notion with the property \mathrm{P}\mathrm{r}_{k+1},|\vdash_{\mathb {P}_{*}^{k}*\mathb {Q}there are no directed subset G of \dot{\mathb {Q} _{*}^{k} such that \dot{I}_{$\alpha$} \cap G \neq \emptyset for all$\alpha$\in$\omega$_{1}This implies that, for any integer k\geq 2 and any \mathb {P}_{*}^{k} ‐name \dot{\mathb {Q} for a forcing notion
with the property \mathrm{P}\mathrm{r}_{k+1},|\vdash_{\mathb {P}_{*}^{k}*\dot{\mathb {Q} “ \mathrm{M}\mathrm{A}_{\mathrm{N}_{1} ( $\sigma$-k ‐linked) fails”’
Therefore, the following has been concluded.

Theorem 1.6 ([2, LEMMA 6\mathrm{M}\mathrm{A}_{\aleph_{1} ( $\sigma$-k ‐linked) fails.

It is consistent that MA (\mathrm{P}\mathrm{r}_{k+1}) holds, and

2. THE RECTANGLE REFINING PROPERTY AND THE MAIN RESULT

Larson‐Todorčevič introduced a property of ccc partitions on [$\omega$_{1}]^{2} , called the rec‐tangle refining property, and obtained a consistency of the affirmative of Katětov’s
problem [6]. The following is a version of the rectangle refining property for forcing

notions. A similar definition of the following is appeared in [9, 11]. The following
notation is inspired by [3, Theorem 3.1].
Definition 2.1. A forcing notion \mathb {P} satisfies the rectangle refining property if there
exists a function w from \mathb {P} into [$\omega$_{1}]^{<\aleph_{0} such thatfor any pair of compatible conditions p and q in \mathb {P} , there exists a common
extension r of p and q in \mathb {P} such that w(r)=w(p)\cup w(q) , and(rec) for any uncountable subsets I and J of \mathb {P} , if the set \{w(p);p\in I\cup J\}
forms a $\Delta$ ‐system, then there are uncountable subsets I' and J' of I and
J respectively such that each element of I' is compatible with any elementof J' in \mathbb{P}.

The rectangle refining property is a stronger property than the countable chain
condition. Like the property \mathrm{P}\mathrm{r}_{k} , typical examples of forcing notions with therectangle refining property are a $\sigma$‐centered forcing notion and a specializationof an Aronszajn tree by finite approximations. For other examples, see [9, 10, 11].
Note that forcing notions with the rectangle refining property satisfies Chodounský‐Zapletal’s Y‐cc [3]. Let\langle\mathb {P}_{$\alpha$}, \dot{\mathb{Q}_{$\beta$} : $\alpha$\leq $\lambda$, $\beta$< $\lambda$\rangle be a finite support iteration offorcing notions with the rectangle refining property, and, for each $\beta$< $\lambda$ , let \dot{w}_{ $\beta$} be\mathrm{a}\mathb {P}_{ $\beta$} ‐name for a function that witnesses the rectangle refining property of \dot{\mathb {Q}_{$\beta$} . By
induction on $\alpha$\leq $\lambda$ , it can be prove that, for every p\in \mathbb{P}_{ $\alpha$} , there is an extension qof p in \mathb {P}_{$\alpha$} such that, for each $\xi$\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(q) , there exists w_{ $\xi$}^{q}\in [$\omega$_{1}]^{<\aleph_{0} such thatq \mathrm{r} $\xi$|\vdash_{\mathbb{P}_{ $\xi$} \dot{w}_{ $\xi$}(q( $\xi$) =w_{ $\xi$}^{q}”Such a q is here called a nice condition of \mathb {P}_{ $\alpha$} . Note that the set of nice conditions of\mathb {P}_{ $\alpha$} is dense in \mathb {P}_{ $\alpha$} . We say that an uncountable set \{p_{ $\zeta$} : $\zeta$\in$\omega$_{1}\} of nice conditionsof \mathb {P}_{$\alpha$} forms a \mathrm{A} ‐system as a set of conditions of the iteration if \{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p_{ $\zeta$}) : $\zeta$\in$\omega$_{1}\}forms a \mathrm{A} ‐system with root \triangle and, for each $\xi$\in\triangle,\{w_{ $\xi$}^{p_{ $\zeta$} : $\zeta$\in$\omega$_{1}\} also forms a\triangle-

system. Note that every uncountable set of nice conditions of \mathb {P}_{$\alpha$} has an uncountable
subset that forms a \triangle ‐system as a set of conditions of the iteration. The rectangle
refining property is preserved under finite support iterations in the following sense.

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Lemma 2.2. Suppose that\langle\mathb {P}_{$\alpha$}, \dot{\mathb{Q}_{$\beta$} : $\alpha$\leq $\lambda$, $\beta$< $\lambda$\rangle is a finite support iteration offorcing notions with the rectangle refining property, and let $\alpha$\leq $\lambda$ . Then, for every
uncountable set \{p_{ $\zeta$} : $\zeta$\in$\omega$_{1}\} of nice conditions of\mathbb{P}_{ $\alpha$} , and every pair of uncountable
subsets I and J of $\omega$_{1} , if \{p_{ $\zeta$} : $\zeta$\in$\omega$_{1}\} forms a $\Delta$ ‐system as a set of conditions of
the iteration, then there are I'\in[I]^{\aleph_{1}} and J'\in[J]^{\aleph_{1}} such that, for each $\zeta$\in I and$\eta$\in J', p_{ $\zeta$} and p_{ $\eta$} are compatible in \mathb {P}_{ $\alpha$}.
Proof. This is proved by induction on $\alpha$\leq $\lambda$ . In the case that $\alpha$ is a limit ordinal,
this is proved by inductive hypothesis. Suppose that \mathbb{P}_{a} satisfies the conclusion of
the lemma, that \{p_{ $\zeta$} : $\zeta$\in$\omega$_{1}\} is an uncountable set of nice conditions of \mathbb{P}_{ $\alpha$+1} suchthat \{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p_{ $\zeta$}): $\zeta$\in$\omega$_{1}\} forms a $\Delta$ ‐system with root \triangle with $\alpha$\in\triangle , and, for eachalso forms a \triangle ‐system, and that and J are uncountable
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By inductive hypothesis, there are I^{(1)} \in [I]^{\aleph_{1} and J^{(1)} \in [J]^{\aleph_{1} such that, foreach $\zeta$ \in I^{(1)} and $\eta$ \in J^{(1)}, p_{ $\zeta$} \mathrm{r} $\alpha$ and p_{ $\eta$} \lceil $\alpha$ are compatible in \mathb {P}_{$\alpha$} . By refiningI^{(1)} and J^{(1)} if necessary, we may assume that I^{(1)} is disjoint from J^{(1)} . Define\mathb {P}_{ $\alpha$} ‐names Ì(2) and j^{(2)} such that|\vdash_{\mathb {P}_{$\alpha$} “ \dot{I}^{(2)}:=\{ $\zeta$\in I^{(1)} p_{$\zeta$} [a\in\dot{G}_{\mathb {P}_{ $\alpha$} \}:and j^{(2)}:=\{ $\zeta$\in J^{(1)} : p_{ $\zeta$} \mathrm{r}a\in\dot{G}_{\mathb {P}_{ $\alpha$} \},,Claim that there exists q\in \mathbb{P}_{ $\alpha$} such that

q1\vdash_{\mathrm{p}_{ $\alpha$} “ both \dot{I}^{(2)} and j^{(2)} are uncountable”
To see this, assume not. Then, since \mathb {P}_{$\alpha$} is ccc, there is $\delta$\in$\omega$_{1} such that|\vdash_{\mathbb{P}_{ $\alpha$}} j^{(2)}, j^{(2)} \subseteq $\delta$”

Take $\zeta$ \in I^{(1)}\backslash $\delta$ and $\eta$ \in J^{(1)}\backslash $\delta$ , and take a common extension r of p_{ $\zeta$} \mathrm{r} $\alpha$ andp_{ $\eta$} \mathrm{r} $\alpha$ in \mathb {P}_{ $\alpha$} . Thenr|\vdash_{\mathbb{P}_{ $\alpha$} ( $\zeta$\in\dot{I}^{(2)}\backslash $\delta$ and $\eta$\in j^{(2)}\backslash $\delta$
which is a contradiction.

\mathb {P}_{ $\alpha$} be a condition of \mathb {P}_{ $\alpha$} that forces \dot{I}^{(2)} and j^{(2)} to be uncountable.
Since \dot{\mathb {Q} _{$\alpha$} is a \mathb {P}_{$\alpha$} ‐name for a forcing notion with the rectangle refining property,
there are \mathb {P}_{ $\alpha$} ‐names \dot{I}^{(3)} and j^{(3)} such that
Let q

\in

q|\vdash_{\mathrm{p}_{ $\alpha$} “ \dot{I}^{(3)} and j^{(3)} are uncountable subsets of \dot{I}^{(2)} and j^{(2)} respectively,
and, for each $\zeta$ \in \dot{I}^{(3)} and $\eta$ \in j^{(3)}, p_{ $\zeta$}( $\alpha$) and p_{ $\eta$}( $\alpha$) are compatible in\dot{\mathb {Q} _{ $\alpha$},For each i\in$\omega$_{1} , take an extension q_{i} of q in \mathb {P}_{$\alpha$} and $\zeta$_{i}, $\eta$_{i} \in$\omega$_{1} such that, for eachi\in$\omega$_{1}, q_{i} forms a nice condition and

q_{i}1\vdash_{\mathb {P}_{ $\alpha$} “ $\zeta$_{i}\in\dot{I}^{(3)} and $\eta$_{i}\in j^{(3)} ,and, for each i, j\in$\omega$_{1} with i<j,\displaystyle \max\{$\zeta$_{i}, $\eta$_{i}\}<\min\{$\zeta$_{j}, $\eta$_{j}\}.
Take an uncountable subset K of $\omega$_{1} such that \{q_{i}:i\in K\} forms a \triangle ‐system asa set of conditions of the iteration. By inductive hypothesis, take uncountabledisjoint subsets K_{0} and K_{1} of K such that, for each i\in K_{0} and j\in K_{1}, q_{i} and q_{j}are compatible in \mathb {P}_{$\alpha$} . Define I' :=\{$\zeta$_{i}:i\in K_{0}\} and J' :=\{$\eta$_{j}:j\in K_{1}\} . Sinceq_{i}^{1\vdash}\mathbb{P}_{ $\alpha$} $\zeta$_{i}\in\dot{I}^{(3)} \subseteq\dot{I}^{(2)}\subseteq I^{(1)}\subseteq I”

66

for each i \in K_{0}, I' is an uncountable subset of I . Similarly, J' \subseteq

J.

The pair I'

and J' is what we want.

\squareLemma 2.3. Suppose that k is an integer not smaller than 2, and \dot{\mathb {Q} is a \mathb {P}_{*}^{k} ‐namefor a forcing notion with the rectangle refining property. Then|\vdash_{\mathb {P}_{*}^{k}*\dot{\mathb {Q} “ there are no K\in[$\omega$_{1}]^{\aleph_{1} such that [K]^{k+1} \subseteq\dot{\mathb {Q} _{*}^{k}This lemma implies that \mathb {P}_{*}^{k}*\dot{\mathb {Q} forces the failure of \mathrm{M}\mathrm{A}_{\mathrm{N}_{1} ( $\sigma$-k ‐linked) for \dot{\mathb {Q} _{*}^{k} insome strong sense. See in the next section.
Proof. Throughout the proof, we work in the extension with \mathb {P}_{*}^{k} . Suppose that \mathb {Q}is a forcing \mathrm{n} otion with the rectangle refining property (in the extension with \mathb {P}_{*}^{k} ),q\in \mathbb{Q} and K is a \mathb {Q}‐name for an uncountable subset of $\omega$_{1} such thatq^{1\vdash}\mathb {Q} [\dot{K}]^{k+1}\subseteq \mathb {Q}_{*}^{k}”

For each $\alpha$ \in$\omega$_{1} , take an extension q_{ $\alpha$} of q in \mathb {Q} and $\delta$_{$\alpha$} \in$\omega$_{1} such that, for each$\alpha$\in$\omega$_{1}, q_{ $\alpha$} forms a nice condition andq_{ $\alpha$}|\vdash_{\mathrm{p}_{ $\alpha$} $\delta$_{ $\alpha$}\in\dot{K} ” ,and, for each $\alpha$, $\alpha$'\in$\omega$_{1} with $\alpha$<$\alpha$', $\delta$_{$\alpha$} <$\delta$_{$\alpha$'} . Note that each set \{$\delta$_{ $\alpha$}\} is a condi‐tion of \mathb {Q}_{*}^{k} . Since \mathb {Q} satisfies the rectangle refining property, we can take uncount‐able subsets I_{l}, l \leqk,of $\omega$_{1} such that, for each \langle$\alpha$_{l} : l\leq k\rangle \in\displayst le\prod_{t\leqk}I_{l}, \{q_{$\alpha$_{ $\iota$} : l\leq k\}

has a common extension in \mathb {Q} . We build a pairwise disjoint uncountable family\{u_{ $\xi$} : $\xi$\in$\omega$_{1}\} of finite subsets of $\omega$_{1} such that each u_{ $\xi$} contains some member of
Applying \{\{$\delta$_{ $\alpha$}\}: $\alpha$\in$\omega$_{1}\} and \{u_{ $\xi$} : $\xi$\in$\omega$_{1}\} to [2, LEMMA 4 (5)](which is a property of \mathb {Q}_{*}^{k} , mentioned above), we can find \{$\alpha$_{l} : l\leq k\rangle
such

I_{l} for all l \leq k .

\displaystyle\in\prod_{l\eqk}I_{l}that \{$\delta$_{$\alpha$_{ $\iota$}} : l\leq k\} \not\in \mathb {Q}_{*}^{k} . However, \{q_{$\alpha$_{l}} : l\leq k\} has a common extension r in \mathbb{Q},and then,r|\vdash_{\mathb {Q} \{$\delta$_{ $\alpha \iota$} :l\leq k\}\in [\dot{K}]^{k+1}\subseteq \mathb {Q}_{*}^{k}\squarewhich is a contradiction.

Therefore, we obtain the following theorem

Theorem 2.4. For each integer k \geq 2 , it is consistent that MA(rec) holds, and\mathrm{M}\mathrm{A}_{\aleph_{1} ( $\sigma$-k ‐linked) fails.
3. CONCLUDING REMARK

Connections between several fragments of Martin’s axiom have been studied.

For example, Bagaria proved that it is consistent that MA( $\sigma$ ‐centered) holds andMA( $\sigma$ ‐linked) fails [1, 3.6], and that it is consistent that MA(productive ccc) holds,
every Aronszajn tree is special and MA fails. Chodounský‐Zapletal introduced
the property of forcing notions, called Y‐cc, which is a stronger property than the
countable chain condition, and showed that it is consistent that MA(Y‐cc) holds and\aleph_{1} [3] . Note that MA(Y‐cc) implies MA ( $\sigma$ ‐centered) and the assertion\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{N})=67

that every Aronszajn tree is special 1. Hence, this Chodounský‐Zapletal’s result
also implies the consistency of the assertion (^{*})^{*2}.
*

In 1980\mathrm{s} , Todorčevič investigated Mai in’s Axiom from the view point of Ramsey
theory, and introduced the following fragments of Martin’s axiom: \mathcal{K}_{< $\omega$} denotes theassertion that every ccc forcing notion has precaliber \aleph_{1} (that is, every uncountable
subset has a centered subset I , which means that any finite subset of I has a common
extension); \mathcal{K}_{n} denotes the assertion that every ccc forcing notion has the property
K_{n} (that is, every uncountable subset of a ccc forcing notion has an uncountable‐linked subset I , which means that any nrmany elements of I has a common
extension) *3_{;\mathcal{K}_{< $\omega$}' denotes the assertion that every ccc partition K_{0}\cup K_{1}=[$\omega$_{1}]^{<\aleph_{0}}has an uncountable K_{0} ‐homogeneous set, for each n\in $\omega$;\mathcal{K}_{n}' denotes the assertion*4
that every ccc partition K_{0}\cup K_{1}=[$\omega$_{1}]^{n} has an uncountable K_{0} ‐homogeneous set.
The following diagram is a summary of implications of these fragments of \mathrm{M}\mathrm{A}_{\aleph_{1} .
$\eta$_{ $\Gamma$}

The triangle on the left side of the diagram is the Todorčevič‐Veličkovič theorem[8].
\rightarrow \mathcal{K}_{n+1}\rightarrow \mathcal{K}_{n}\rightarrow\rightarrow \mathcal{K}_{3}\rightarrow \mathcal{K}_{2}\mathrm{M}\mathrm{A}_\aleph_{1}<_{\mathrm{A}<^{> \mathcl{K}_<$\omega$} \uparow\rightarow. \rightarow\mathcal{K}_{n+1}'\downarow\rightarow\mathcal{K}_{$\tau\iota$}\downarow,\rightarow \rightarow\mathcal{K}_{3|,\rightarow\mathcal{K}_{2'\downarow\mathcal{K}_{< $\omega$}'\rightar ow

.

.

.

.

.

.

.

.

..

.

It is not known whether any other implications in this diagram hold under ZFC.

Bagaria‐Shelah’s lemma [2, LEMMA 6] can be modified the lemma for the failureof \mathcal{K}_{k+1}' for \dot{\mathb {Q} _{*}^{k} in the extension with \mathb {P}_{*}^{k}*\dot{\mathb {Q} . So it is proved that, for each integer
k\geq 2 , it is consistent that MA (\mathrm{P}\mathrm{r}_{k+1}) holds and \mathcal{K}_{k+1}' fails.

Larson‐Todorčevič showed that a Suslin tree forces that there exists a laddersystem coloring which cannot be uniformized [5, THEOREM 6.2], and that, for each
non‐principal ultrafilter U in the ground model, (2^{$\omega$_{1} , <\mathrm{l}\mathrm{e}\mathrm{x}) cannot be embedded
into $\omega$^{ $\omega$}/U [ 5 , THEOREM 6.3]. It is proved that \mathcal{K}_{4}' implies that every ladder system
coloring can be uniformized [8, §2], and that \mathcal{K}_{3}' implies that, for every non‐principal
ultrafilter U in the ground model, (2^{$\omega$_{1} , <\mathrm{l}\mathrm{e}\mathrm{x}) can be embedded into $\omega$^{ $\omega$}/U [7 , 7.7.
THEOREM]. Larson‐Todorčevié proved that it is consistent that a Suslin tree canforce \mathcal{K}_{2}' (rec) [6]. In [11], the author develops their result to \mathcal{K}_{< $\omega$}(\mathrm{r}\mathrm{e}\mathrm{c}) in somesense. Therefore, it is proved that it is consistent that \mathcal{K}_{<\mathrm{I}v}(\mathrm{r}\mathrm{e}\mathrm{c}) holds in somesense and \mathcal{K}_{3}' fails, by use of forcing with a Suslin tree. Lemma 2.3 says that\mathcal{K}_{k+1}' for \dot{\mathb {Q} _{*}^{k} fails in the extension with \mathb {P}_{*}^{k}*\dot{\mathb {Q} . So consequently, it is proved that

it is consistent that MA(rec) holds, and both \mathcal{K}_{3}' and \mathrm{M}\mathrm{A}_{\aleph_{1} ( $\sigma$‐linked) fail. Thiscannot be concluded by use of a forcing extension with a Suslin tree.

”Because both a $\sigma$ ‐centered forcing and a specialization of an Aronszajn tree by finite approx‐
imations satisfy Y‐cc.
*2
Notice that Random forcing is $\sigma$ ‐linked.
*3\mathrm{A} forcing notion with the property K_{n} satisfies the property \mathrm{P}\mathrm{r}_{n}.
*4They are defined by Todorčevič in several papers. In [5, Definition4.9] and [8, §2], \mathcal{K}_{n} ’s aredefined as assertions for ccc forcing notions, however in [6, §4] and [7, §7], \mathcal{K}_{n} ’s are defined asassertions for ccc partitions. To separate them, we use the notations as above. These notations

are same to ones in [10].

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FACULTY 0F SCIENCE, SHIZUOKA UNIVERSITY, OHYA 836, SHIZUOKA, 422‐8529, JAPAN.
E ‐mail address: yorioka@shizuoka.ac.jp