数理解析研究所講究録
第 1595 巻 2008 年 20-31

20

On pair-splitting and pair-reaping pairs of$\omega$

Hiroaki Minami
Abstract
In this paper we investigate variations of splitting number and, pair-reaping number
reaping number, pair-splitting number.
. We also prove it is conWe prove that it is consistent that.
sistent that$\mathfrak{s}_{pair}$$\mathfrak{r}_{pa1r}$$\mathfrak{s}_{pair}<\mathfrak{d}$$\mathfrak{r}_{pair}>b$

Introduction
The splitting number and the reaping number are cardinal invariants.
related to the structure
we say $X$ splits if $X\cap Y$ andare infinite. We
For $X,$, there exists
a splitting family if for eachcallsuch that $X$ splits Y. The splitting number is the least size of a splitting
family.
a reaping family if for each $X\in[\omega]$ , there exists
We call
is finite. The
such that is not split by $X$ , that is, $X\cap Y$ is finite orreaping number is the least size of a reaping family.
We shall study variations of splitting number and reaping number, pair. They are introduced
and pair-reaping number
splitting number
and investigated in [7] to analyze dual-reaping number and dual-splitting
number which are reaping number and splitting number for the structure ofrespectively.
all infinite partitions of ordered by “almost coarser”$a\in
A$ such that, there exists
unbounded if forWe$X$ pair-splits
if there. For
and unbounded. We callandexist infinitely many $a\in A$ such that, there exists
a pair-splitting farnily if for each unbounded$X\in S$ such that $X$ pair-splits. The pair-splitting number
is the least
size of a pair-splitting family.
$z$$\mathfrak{r}$$\mathcal{P}(\omega)/fin$$Y$$Y\in[\omega]^{w}$$Y\backslash X$$Y\in[\omega]^{w}$$S\subset[\omega]^{\omega}$$X\in[\omega]^{w}$$\mathfrak{s}$$Y\in[\omega]^{\omega}$$\mathcal{R}$$Y$$Y\backslash X$$\mathfrak{r}$$\mathfrak{r}_{pair}$$\mathfrak{s}_{pair}$$\mathfrak{r}_{d}$$\mathfrak{s}_{d}$$((\omega)^{w}, \leq^{*})$$\omega$$k\in\omega$$caJlA\subset[\omega]^{2}$$a\cap k=\emptyset$$A$$A\subset[\omega]^{2},$$X\in[\omega]^{\omega}$$a\cap X\neq\emptyset$$a\backslash X\neq\emptyset$$A\subset[\omega]^{2}$$S\subset[\omega]^{w}$$A$$\ovalbox{\tt\small REJECT}_{pair}$

21

isa pair-reaping family if for each
We call$X$
doesn’t pair, there exists
such that
unbounded and foris the least size of a pair-reaping
split . The pair-reaping number
family.
,In [7] it is proved that there is the following relationship between
and other cardinal invariants.
$A\in \mathcal{R},$$\mathcal{R}\subset \mathcal{P}([\omega]^{2})$$A$$A\in \mathcal{R}$$X\in[\omega]^{w}$$A$$r_{pair}$$t_{pair}$$\mathfrak{s}_{pair}$1.

Proposition 0.1

2.
3.

4.

$\mathfrak{r}_{pa\dot{j}r}\geq cov(\mathcal{M}),$$\mathcal{B}_{pair}\geq \mathfrak{s}$$\mathfrak{s}_{pair}\leq non(\mathcal{M}),$ $non(\mathcal{N})$$cov(N)$.

.

.

$\mathfrak{r}_{pair}\leq \mathfrak{r},\mathfrak{s}_{d}$.

It is not known that
Question 0.1$\mathfrak{r}_{d}\leq \mathfrak{s}_{pair}$$\mathfrak{r}_{d}\leq z_{pair}$or not.

?

andstatement in [7]:hold (see in [2]). And Kamo proved the following

Theorem 0.1

and$\mathfrak{s}\leq \mathfrak{d}$$\mathfrak{r}\geq b$$\mathfrak{r}_{d}\leq \mathfrak{d}$$s_{d}\geq b$.

So we have the following diagram:

22

In [7] by using finite support iteration of Hechler forcing, the following
consistency results are proved.

Theorem 0.2 It is consistent that $z_{\mu ir}<add(\mathcal{M})$ . Dually it is consistent).
that$\mathfrak{r}_{\mu ir}>co\lambda \mathcal{M}$is a lower bound of and andis an upper bound of (and maybe). So it is natural to ask the following question.
$\mathfrak{r}$$\mathfrak{r}_{pair}$of$\mathfrak{r}_{d}$Question 0.2$\mathfrak{s}_{pair}\leq \mathfrak{d}$$z$$\mathfrak{s}$$\mathfrak{s}_{pa1r}$? Dually$\mathfrak{r}_{pair}\geq b$?

In the present paper we shall investigate the relation ship between
andand the relationship between
and D. In section 1 we shffi prove the. In section 2 we shall show the consistency of theconsistency of. In section 3 we mention the development of results
consistency of
in section 1 and 2.
$\mathfrak{r}_{pair}$$b$$\mathfrak{s}_{pair}$$\mathfrak{s}_{pair}>\mathfrak{d}$$\mathfrak{r}_{pair}<b$231pair-splitting number and dominating number
Notation and Definition We present the related notions. We use standard
denotes the set of
set theoretical conventions and notation. For a set $X,$
all functions ffom to $X$ . For $f,g\in\omega^{w},$ dominates , written $f\leq^{*}g$ , if
for all but finitely many $n\in\omega g(n)\leq f(n)$ . We call a dominating ftlily
such that $g\leq*f$ . The dominating
there exists
if for eachnumber is the least size of a dominating family.
suChthere exists
We call an unbounded family if for eachsuch that $g(n)>f(n)$ .
that $g\not\leq*f$ , i.e., there exist infinitely many
The unbounded number is the least size of an unbounded family.
For a set $x,$ $x<w$ denote the set of all functions from naturd numbers toX.
We call partial ordering $(T, <)$ a tree if the set $\{s\in T : s<t\}$ isordered by . We say is a tree on $X$ if is a subtree of $(x<w\subset)$ . For ais the set of all immediate successors of in .
tree and $t\in T,$
For a tree $T,$ $stem(T)$ is the first element of which has at least 2-many
immediate successors.
$X^{w}$$f$$\omega$$g$$\mathcal{F}$$g\in\omega^{w}$$f\in \mathcal{F}$$0$$g\in \mathcal{G}$$f\in\omega^{w}$$\mathcal{G}$$n\in\omega$

$b$

wen-

$T$

$<$

$T$

$T$

$t$

$su\omega_{T}(t)$

$T$

$T$

Theorem 1.1 It is consistent$\mathfrak{s}_{pair}>\mathfrak{d}$.

To prove theorem 1.1, we shall construct a proper forcing notion which
and is -bounding to show is preserved by the forcing notion.
enlarges$\omega^{w}$$\mathfrak{d}$$\mathfrak{s}_{pair}$Definition 1.1 /4 pp3407 A forcing notion$\mathbb{P}$is$\omega^{w}$-bounding if$\mathfrak{l}\vdash r\forall f\in\omega^{w}\cap V[G]\exists g\in\omega^{\omega}\cap V(f\leq*g)$The$\omega^{w_{-}}bound_{\dot{i}}$.

gness has the following good property.

Theorem 1.2 /4 ppS41] The countable support iteration of prope-bounding.
forcing notions is$r\omega^{\omega}$-bounding$\omega^{w}$

To prove theorem 1.1 we shall construct a forcing notion which consists
such that the set of successors of any nodeof finitely branching trees oncarries a norm as [8].
To present the desired forcing notion, we define “norm” for finite subsets. Let $R(n)$ be a natural number such that if $m\geq R(n)$ , then for anyof$[\omega]^{2}$$[\omega]^{2}$24function : $[m]^{2}arrow 2$ there exists $H\in[m]^{n}$ such that $|f([H]^{2})|=1$ . Thenrecursively define $l_{1}=3,$ $l_{n+1}= \max\{2l_{n}, R(l_{n})\}$ . Then for a finite subset
of $[\omega]^{2}norm(A)\geq n$ if contains a complete graph with -many vertices.
This norm has the following properties:
$f$

$A$

$A$

$l_{n}$

Proposition 1.1 For a finite subset

1. no $rm(A)\geq 1implie8$ for any$a\cap X=\emptyset$

or$a\subset X$

$A$

of

$[\omega]^{2}$,
there exists $a\in A$ such that

$X\in[\omega]^{w}$.

2. Suppose norm$(A)\geq n+1$ . For
let $A_{X}^{0}=\{a\in A:a\cap X=\emptyset\}$and $A_{X}^{1}=\{a\in A:a\subset X\}$ . Then nom
or norm.
$X\in[\omega]^{w}$$(A_{X}^{0})\geq n$$(A_{X}^{1})\geq n$

3. Suppose norm$(A)\geq n+1$ . If $A=A_{0}\cup A_{1}$ , then no $7m(A_{0})\geq n$ ornorm $(A_{1})\geq n$ .
Proof of proposition 1.1
1. Since $n\sigma rm(A)\geq 1,$ contains a complete graph $A’\subset A$ with 3-many
vertices. Then for any 2-coloring of the vertices of $A’$ , there exists an edge
whose vertices have the same color. So there exists $a\in A’\subset A$ such that$A$

$a\subset X$

or$a\cap X=\emptyset$.

contain a complete graph $A’$ with
2. Since norm $(A)\geq n+1,$$X$ contains
many vertices. So for eacb
-many vertices of $A’$
or $X$ doesn’t meet -many vertices of $A’$ because $l_{n+1}\geq 2l_{n}$ . Anyway$A_{X}^{0}=\{a\in A:a\cap X=\emptyset\}$ or $A_{X}^{1}=\{a\in A:a\subset X\}$ contains a comPlete
graPh with -many vertices. Therefore norm
or norm
.
$A’$
-many
3. Since norm $(A)\geq n+1,$ contain a complete graph with
for $i<2$ . Since $l_{n+1}\geq R(l_{n})$ ,vertices. Define : $A’arrow 2$ by $f(a)=i$ ifwhich has -many vertices of $A’$ andthere exists a complete graph$|f[A^{*}]|=1$ . Soor
. Hence norm $(A_{0})\geq n$ or norm $(A_{1})\geq n$ .
$A$$l_{n+1^{-}}$$X\subset\omega,$$l_{n}$$l_{n}$$(A_{X}^{0})\geq n$$l_{n}$$A$$(A_{X}^{1})\geq n$$l_{n+1}$$a\in A_{i}$$f$$A^{*}\subset A’$$l_{n}$$A^{*}\subset A_{1}$$A^{*}\subset A_{0}$口Then let1.

$T$$\mathbb{P}$be the set of perfect trees such thatis a finitely branching tree on$[\omega]^{2}$,

there exist $m\geq n$ such that whenever
2. for any branch of and$t\in T$ with $|t|\geq m,$ $norm(succ_{T}(t))\geq n$ .
$T$

For $T\bm{t}dS$ in$\mathbb{P},$$T\leq S$$n\in\omega$if$T\subset S$.

25

Lemma 1.1 Let be a generic filter on and $A_{G}=\cap\{T:T\in G\}$ . Then.
and for any $X\in[\omega]^{w}nV,$ $X$ doesn’t pair-split$G$$\mathbb{P}$$A_{G}$$A_{G}\subset[\omega]^{2}$of by $T\in D_{X}$ if for all $t\in$Proof For $X\in[\omega]^{w}$ define a subset$T\backslash \{s : s\subset stem(T)\}$ and $a\in succ_{T}(t),$ $a\subset X$ or $a\cap X=\emptyset$ .
Then for a
given
we can find $T\leq S$ such that for all $t\in T\backslash \{s:s\subset stem(T)\}$and $a\in succ_{T}(t),$ $a\subset X$ or $a\cap X=\emptyset$ by 1 and 2 in Proposition 1.1. So.
is dense. So $X$ doesn’t pair-split$D_{X}$$\mathbb{P}$$S\in \mathbb{P}$$D_{X}$$A_{G}$口which is not pair-split
By this lemma, adds an infinite subset ofby any infinite subset of in ground model. Therefore -stage countable.
support iteration of forces
Fkom now on we shall prove is -bounding and proper.
$T\leq*S$ if, let $ess(T)=\{t\in T:stem(T)\subset t\}$ . For $T,$For
$T\leq S$ and for all $t\in ess(T),$ $norm(succ_{T}(t))\geq norm(succ_{S}(t))-1$ . $T\leq_{m}S$
if $T\leq S$ and for all $t\in T$ with norm $(succ_{S}(t))\leq m$ , we have $succ_{S}(t)\subset T$ .
As [8] we can prove the following lemmata.
$[\omega]^{2}$$\mathbb{P}$$\omega$$\omega_{2}$$\mathbb{P}$$\mathfrak{s}_{pair}=\omega_{2}$$\mathbb{P}$$\omega^{w}$$S\in \mathbb{P},$$T\in \mathbb{P}$

Lemma 1.2 If$S\in \mathbb{P}$I. every branch of
II.

$T$

and $W\subset S$ , then there is some $T\leq^{*}S$ such that$T$

meets $W$ , or else

is disjoint from $W$ .

such
be the set of all $s\in S$ such that there exists
Proof Let$S’$is the set of $t\in S$ comparable tomeets $W$ wherethat every branch of$S’\leq^{*}S_{\delta}$$S^{W}$$S_{8}$$s$.

If stem $(S)\in S^{W}$ , then (I) holds. Otherwise we will construct $T\leq*S$
which satisfies (II).
Suppose stem $(S)\not\in S^{W}$ . Recursively construct $t\in T$ with $|t|=n$ . If
$n\leq|stem(T)|,$ $t\in T$ with $|t|=n$ if $t\in S$ with $|t|=n$ . If $n\geq|stem(T)|$ ,for $t\in T$ with $|t|\leq n$ .
assume $t\in T$ with $|t|\leq n$ are given and.
and
For $t\in T$ with $|t|=n$ , let $A^{t}=succ_{S}(t),$By Proposition 1.1 (iii), norm $(A_{i}^{t})\geq norm(A^{t})-1$ for some $i<2$ . Since
holds I. So norm $(A_{0}^{t})<n$ . Hencesuch that, there is nonorm $(A_{1}^{t})\geq norm(A^{t})-1$ . Define $t\in T$ with $|t|=n+1$ if $t\lceil n\in T$ and.
. Then for any $t\in T$ with $|t|=n+1,$
By construction $T\leq^{*}S$ and satisfies II.
$t\not\in S^{W}$$A_{0}^{t}=S^{W}\cap A^{t}$$t\not\in S^{W}$$t(n)\in A_{1}^{trn}$$S’\leq^{*}S_{t}$$A_{1}^{t}=A^{t}\backslash A_{0}^{t}$$S’$$t\not\in S^{W}$$\square$26

Lemma 1.3 Let

be a -name for an ordinal. Let $S\in \mathbb{P}8uch$ that for$t\in S\backslash \{s:s\subset stem(S)\},$ $norm(succ_{S}(t))>m+1$ .
Then there exists$T\leq_{m}S$ and a.
finite subset of ordinal such that$\dot{\alpha}$$\mathbb{P}$$T|\vdash\dot{\alpha}\in w$$w$.

Proof Let $W$ be the set of nodes $s\in S$ such that there exists $S^{S}\leq_{m}S$
which decides the value 6.
We shall prove that there exists $S_{1}\leq*S$ such that every branch ofmeets $W$ . Suppose $S’\leq*S$ and $S”\leq S’$ such thatforsome . Then for some $t\in S’’$ for each extension of in $S”$ satisfies$t\in W$ . Hencenorm $(succ_{S’’}(s))>m$ . Because
and $S”$ decides
by Lemma 1.2 there exists
which satisfies I in Lemma 1.2.
Let $S_{1}\leq*S$ such that every branch ofmeets $W$ . Let
be theset of minimal elements of $W$ in . Since
is finitely branching,is(Otherwise,
K\"oning’s
byfinite.
Lemma we can construct infinitely branch
$W$
$v\in
W_{0}$ choose
which doesn’t meet ). For
andsuch that$w=\{\alpha_{v}:v\in
W_{0}\}$. Putand. Then $T\leq_{m}S$.
and$S_{1}$$S”|\vdash\dot{\alpha}=\beta$$\beta$$s$$S_{t}’’\leq_{m}S_{t}$$t$$\dot{\alpha},$$S_{1}\leq^{*}S$$S_{1}$$S_{1}$$W_{0}$$S_{1}$$W_{0}$$T^{v}\leq_{m}S_{v}$$T^{v}|\vdash\dot{\alpha}=\alpha_{v}$$\alpha_{v}$$T= \bigcup_{v\in W_{0}}T^{v}$$T|\vdash\dot{\alpha}\in w$

口Lemma 1.4 If$exi_{8}tsT\leq_{m}S$$S\in \mathbb{P},\dot{\alpha}$

be a -name for an ordinal and $m<w$ . Then there$\mathbb{P}$and a finite set of ordinals$w$such that$T|\vdash\dot{\alpha}\in w$.

Proof Choosesuch that for any $s\in S$ with $|s|\geq knorm(succ_{S}(s))>$$m+1$ . For each $s\in S$ with $|s|=k$ , apply Lemma 1.3 topickand a finite set of ordinals
so that
. Put
and
and
. Since isbranching, is a finite set.
Proof of theorem 1.1 Lemma 1.4 implies that is -bounding. Given a, we can construct a sequence-name for a function from to andof conditions of such that $T_{0}=S,$and for each, there exists some finiteof natural numbers such that $T_{n}|\vdash;(n)\in$. Then there exists
such that $T\leq\tau$ and $T|\vdash\forall n\in\omega(f(n)\in w_{n})$ .
Put $g(n)=masc\{w_{n}\}$ . Then $T1\vdash\forall n\in\omega(f(n)\leq g(n))$ . So is -bounding.
Also this claim say satisfies Baumgartner’s Axiom A. Hence is proper.
Hence the -stage countable support iteration of is-bounding by$V\modelsCH$,
theorem 1.2. Therefore ifthen the -stage countable support
is a dominating family. So the -stage countable
iteration of forces. Hence it is consistent that
support iteration of forces.
$k\in\omega$$T^{\epsilon}\leq_{m}S_{\delta}$$S_{s}$$T_{\epsilon}\mathfrak{l}\vdash\dot{\alpha}\in w_{\epsilon}$$w_{\epsilon}$$T= \bigcup_{\epsilon\in S,|\iota|=k}T_{t}$$T|\vdash\dot{\alpha}\in w$$S$$w_{i}= \bigcup_{\epsilon\in S\cap w^{k}}w_{f}ThenTte\leq_{m}S$$fi_{\dot{P}}te1y\square$$w$$\mathbb{P}$$\dot{f}$$\mathbb{P}$$\langle T_{n} :n\in\omega\rangle$$w$$\omega$$S\in \mathbb{P}$$T_{n+1}\leq_{n}T_{n}$$\mathbb{P}$$n\in\omega$$\omega^{w}$$w_{n}$$T\in \mathbb{P}$$w_{n}$$\mathbb{P}$$\mathbb{P}$$\omega^{w}$$\mathbb{P}$$\omega^{w}$$\mathbb{P}$$\omega_{2}$$\omega_{2}$$\mathbb{P}$$\omega^{w}\cap V$$\omega_{2}$$\mathbb{P}$$\mathfrak{d}=\omega_{1}$$s_{pair}>\mathfrak{d}$$\square$

27

Since

$\mathfrak{s}\leq \mathfrak{d}(see[2])$, we have the following corollary.

Corollary 1.1 It is consistent that$\mathfrak{s}<\mathfrak{s}_{pair}$.

pair-reaping number and unbounded number2, we shall use the Laver forcing L.
To show the consistency ofis a tree and for $s\in T$ with stem $(T)\subset s$ ,is defined by $T\in L$ if
1 $succ_{T}(s)|=\aleph_{0}$ . is ordered by inclusion. Then adds an unbounded real.

$L$$\mathfrak{r}_{pair}<b$$T\subset\omega^{<w}$

$L$

$L$

Proposition 2.1 Let
be a L-generic over $V$ and $f_{G}=\cup\{stem(T)$ : $T\in$$G\}$ .
.
dominates for allThenandis -stage countable support iteration of Laver forcing,
Therefore if
.
then
$G$

$f_{G}\in\omega^{w}$$g\in\omega^{\omega}\cap V$$f_{G}$$L_{w_{2}}$$w_{2}$$V^{L_{v_{2}}}\models b=c$

By using -stage countable support iteration of Laver forcing, we shallconstruct ZFC model which satisfies $r_{pair}<b$ .
$\omega_{2}$Theorem 2.1 It is consistent$\mathfrak{r}_{pair}<b$

By proposition 2.1 it is enoughproperty.

$L$

Definition 2.1 $/4J$ A forcing notion
$H$ :

.

preserves$\mathbb{P}$$\mathfrak{r}_{pair}$. We shall use the Laver

have the Laver property if for $eve\eta$$\omegaarrow\omega\in V$$|\vdash\forall f\in(\Pi_{n\in\omega}H(n))\cap V[\dot{G}]\exists A$: $\omegaarrow\omega^{<w}\in V\forall n\in\omega(f(n)\in A(n)\wedge|A(n)|\leq 2^{n})$Theorem 2.2 /41 The Laver property is preserved under countable support
itemtion of proper forcing notions.

Theorem 2.3 /1, $pp353J$ The Laver forcing

So$L_{w_{2}}$then$\mathbb{P}$$L$

has the Laver property.

has the Laver property. If forcing notion
has the following good property:

Lemma 2.1 Let$\mathbb{P}$$\mathbb{P}$

has the Laver property,

be a forcing notion satisfying the Laver property. Then
( $X$ doesn’t pair-split ).

$|\vdash r\forall\dot{X}\in V[\dot{G}]\exists A\in V$

$A$

28

Proof Let. Let
be an interval partition of suchthat $|I_{n}|=2^{2^{n}}+1$ . Then \langle X:. By the Laver property
there exists $q\leq pp$.such thatsuch that$q|\vdash\forall
n\in\omega(XrI_{n}\in
A_{n})$and. For each-many element. But $|I_{n}|=2^{2^{n}}+1$ . So there exists
is at mostandinsuch that
and. Put
and $A=\{a_{n} : n\in\omega\}\in V$ . Then $q|\vdash XrI_{n}\cap a_{n}=\emptyset$ or$a_{n}\subset XrI_{n}$ for. Therefore
doesn’t pair-split A.
$\Pi=\langle I_{n} :$p\in \mathbb{P}$n\in w\rangle$$fI_{n}$$n\in\omega\rangle$$\langle A_{n} :$\omega$$\in\Pi_{n\in w}2^{I_{n}}$n\in w\rangle\in V$$A_{n}\subset 2^{I_{n}},$$n\in\omega\{\langle\sigma(k) :$|A_{n}|\leq 2^{n}$\sigma\in A_{n}\rangle :k\in A_{n}\}$$2^{2^{n}}$$k_{0}^{n}$$k_{1}^{n}$$I_{n}$$k_{0}^{n}\neq k_{1}^{n}$$\langle\sigma(k_{0}^{n}) :\sigma\in A_{n}\rangle=\langle\sigma(k_{1}^{n}) :\sigma\in A_{n}\rangle$$a_{n}=\{k_{0}^{n}, k_{1}^{n}\}$$q|\vdash\dot{X}$$n\in\omega$$\square$

Proof of theorem 2.1 Suppose $V\models CH$ . By theorem 2.2 and 2.3
has the Laver property. By lemma 2.1 for eachthere$A\subset[w]^{2}$exists an unboundedsuch that
doesn’t Pair-sPlit .
: unbounded}So {is Pair-reaping family. Since $V\models CH$ ,: unbounded}
has the cardinality at most . Therefore{$L_{w_{2}}$$X\in[\omega]^{\omega}\cap V^{L_{w_{2}}}$$A$$V^{L_{w_{2}}}\models X$$A\subset[\omega]^{2}$$A$$\cap V$$\cap V$$A$

$A\subset[w]^{2}$$V^{4_{2}}\models \mathfrak{r}_{pair}<b$$\omega_{1}$

.

口

Since

$\mathfrak{r}\geq b(8ee[2])$, we have the following corollary.

Corollary 2.1 It is consistent that$\mathfrak{r}>\mathfrak{r}_{pair}$.

In [5] Masaru Kada introduces a cardinal invariant as sociated with the Laverproperty.
Let $S$ be the collection of functions from to $[w]<w$ such that $|\phi(n)|\leq$$n+1$ .
is the smallest cardinal such that for every $h\in w^{w}$ there is a set
with cardinality so that, for every $f\in w$ with $f(n)<h(n)$ for all$n<\omega$ , there issuch that for all but finitely many $n\in\omega$ we have$\phi$$\omega$$\mathfrak{l}$$\kappa$$\Phi\subset S$$\kappa$$\phi\in\Phi$$f(n)\in\phi(n)$.

As the proof of theorem 2.1 we can prove the folowing statement.
Corollary 2.2$\mathfrak{r}_{pair}\leq \mathfrak{l}$.

Pawlikowski shows that the dual notion to the definition of is the charac, transitive additivity of null ideal (see [1, pp91]).
terization of trans-add
is the smaJlest size of -bounded family
suchThat is, trankaddthat for every $\phi\in S$ there is $f\in F$ such that for inffiitely many $n\in w$ such$l$$(\mathcal{N})$$(\mathcal{N})$$\leq*$

that $f(n)\not\in\phi(n)$ .
Then the dual inequality to the corollary 2.2 holds.
Proposition 2.2 $z_{pair}\geq tmns- add(\mathcal{N})$ .

$F\subset\omega^{w}$

29

It is known the following relation between trans-add

Theorem 2.4 $/6J$ It is consistent that tmns-add$(\mathcal{N})>\mathfrak{d}$By theorem 2.4 and proposition 2.2 it is consistent that3$(\mathcal{N})$and D.

.

$\mathfrak{s}_{pair}>\mathfrak{d}$.

Further results

In this section we mention the development of above results in the paper [3]written by Hru\v{s}\’ak, Meza-Alc\’antara and the author.
Hru\v{s}\’ak and
Alc\’antara study ctdinal invariants of ideals on andthey deflne the pair-splitting number and the pair-reaping number independently of the author and they showed the pair-splitting number and thepair-reaping number are described as cardinal invariants of an ideal on .
Let be an ideal on . Define the cardinal invariants associate withby
$Meza\ulcorner$$\omega$$w$$\mathcal{I}$$\mathcal{I}$$\omega$$cov^{*}(\mathcal{I})$$= \min\{|\mathcal{A}| :non*(I)$= \min\{|\mathcal{A}| :Theorem 3.1 $/3J$ Let\mathcal{A}\subset \mathcal{I}\wedge\forall I\in \mathcal{I}\exists A\in \mathcal{A}(|A\cap I|=\aleph_{0})\}$.

\mathcal{A}\subset[\omega]^{w}\wedge\forall I\in \mathcal{I}\exists A\in \mathcal{A}(|A\cap I|<\aleph_{0})\}$$\mathcal{G}_{FC}$be an ideal on$[\omega]^{2}$$\mathcal{G}_{FC}=\{A\subset[\omega]^{2} :defined by\chi(w, A)<\aleph_{0}\}$where $\chi(\omega, A)=\min\{k\in\omega : \exists f :warrow k\forall a\in A(|f[a]|=2)\}$ .
.
Then non“and$co\theta(\mathcal{G}_{FC})=\mathfrak{s}_{pair}$$(\mathcal{G}_{FC})=\mathfrak{r}_{pair}$Rom now on we assume is equipped with product topology and the topolis induced by identification of each subset of with its characogy ofteristic function.
. As theorem 2.4, 1.1 and theorem 2.1 weis an -ideal on
Then
can show the following theorem.
$2^{w}$$\mathcal{P}(w)$$\mathcal{G}_{FC}$$\omega$$[\omega]^{2}$$F_{\sigma}$

Theorem 3.2 Suppose$\mathcal{I}$is an$F_{\sigma}$-ideal on .
$\omega$.

1. $/6J$ It is consistent that$\mathfrak{d}<cov^{\triangleright}(\mathcal{I})$2. /3] It is consistent that$b>non^{*}(\mathcal{I})$.

Also the following statement holds as corollary 2.2 and proposition 2.2.

30Corollary 3.1 Suppose$\mathcal{I}$is an

1. If non*(I)\neq w, then non“
2. If non*(I)\neq w, then coif“$F_{\sigma}$-ideal.

$(\mathcal{I})\leq \mathfrak{l}$.

$(\mathcal{I})\geq tmns- add(\mathcal{I})$.

So many results in section 1 and 2 follows bom theorem 3.2 and corollary
3.1.

Acknowledgment
While carrying out the research for this paper, I discussed my work with J\"org
Brendle. He gave me helpful advice. I greatly appreciate his help.
I ako thank Shizuo Kamo for pointing out some remarks. I also thank
Masaru Kada for pointing out corollary 2.2, proposition 2.2 and anotherproof for theorem 2.1 $hom$ proposition 2.2 and theorem2.4.
I thank to Michael Hrugffi and David Meza-Alc\’antara who point out therelation between their results and my research. The collaboration produce
theorem 3.22 and corollary 3.1.
I also thank Teruyuki Yorioka and Noboru Osuga for pointing out somemistake of proof and for suggestions which improved the presentation of thiswork.
Finally I thank members of Arai Project at Kobe University for muchsupport while carrying out the research.

References[1] Tomek Bartoszy\’{n}ski, Haim Judah, “Set theory. On the structure of thereal line”. A K Peters, Ltd., Wellesley, MA, 1995.

[2] Andreas Blass, “Combinatorial cardinal characteristics of the continuum”, in Handbook of Set Theory (A.Kanamori et al., $eds.$ ), $to$ appear.
[3] Michael Hru\v{s}\’ak David Meza-Alc\’entara and Hiroaki Minami, “Aroundpair-splitting and pair-reaping number”, preprint.
[4] Martin Goldstem, “Tools for your forcing construction”. Set theory of thereals (Ramat Gan, 1991), 305-360, Israel Math. Conf. Proc., 6, Bar-Ilan
Univ., Ramat Gan, 1993.

31[5] Masaru Kada, “More on Cicho\’{n}’s diagram and infinite games”, J. Symbolic Logic 65 (2000), no. 4, 1713-1724.
[6] Laflamme, Claude, “Zapping small filters”, Proc. Amer. Math. Soc. 114(1992), no. 2, 535-544.

[7] Hiroaki Minami, “Around splitting and reaping number for partitions of, submitted Aug 2007.
$\omega’$[8] Saharon Shelah, “Vive la diffe’rence. I. Nonisomorphism of ultrapowersof countable models”. Set theory of the continuum (Berkeley, CA, 1989),
357-405, Math. Sci. Res. Inst. Publ., 26, Springer, New York, 1992.