数理解析研究所講究録1095 巻 1999 年 22-27

22Partition properties of subsets ofMASAHIRO SHIOYAA BSTRACT. Let $\kappa>\omega$ be a regular cardinal and$P_{\kappa}\lambda$$\mathrm{R}^{\overline{\prime}}’\Gamma_{P^{\backslash }\frac{\overline{|\Supset}}{\text{ノ}\backslash }fiL_{\backslash }}^{-}\iota/\neg J$a cardinal. The following
partition property is shown to be consistent relative to supercompactness: For any$f$ :withunbounded and $1<\gamma<\kappa$ there is an unboundedwith “for any $n<\omega$ .
$X\subset P_{\kappa}\lambda$$\bigcup_{n<\omega}[\mathrm{x}]_{\subset}^{n}arrow\gamma$$\mathrm{Y}\subset X$

Let$\kappa$$|f$$\lambda>\kappa$$[\mathrm{Y}]_{\subset}^{n}|=1$

be a regular cardinal

properties of the form$>\omega,$$\lambda$$P_{\kappa}\lambdaarrow(F^{+})_{2}^{2}$a cardinal$\geq\kappa$

and $F$ a filter on$P_{\kappa}\lambda$. Partition(see below for the definition) were introduced

by Jech [Je] and successfully used to characterize supercompactness: Menas [Me]proved$P_{\kappa}\lambdaarrow(C_{\kappa\lambda}^{+})_{2}^{2}$for$\kappa 2^{\lambda^{<\kappa}}$-supercompact via a normal ultrafilter on$P_{\kappa}\lambda$

withthe partition property. As noted by Kamo [Kam], Menas’ argumnt can be modified
to derive$P_{\kappa}\lambdaarrow(C_{\kappa\lambda}^{+})_{2}^{2}$

directly from -supercompactness of . For the converse$\lambda$$\kappa$

direction Di Prisco-Zwicker [DZ] and others refined the global result of Magidor[Mag]:$\lambda$-supercompactness of$\kappa$

folows from

$P_{\kappa}2^{\lambda^{<\kappa}}arrow(c_{\kappa 2^{\lambda}}^{+}<\kappa)_{2}2$.

Johnson [Jo] studied properties of the form $Xarrow(F^{+})_{2}^{2}$ for $X\in F^{+}$ , whichmeans that for any$f$:$[X]_{\subset}^{2}arrow 2$

there is$\mathrm{Y}\in F^{+}$

with$\mathrm{Y}\subset X$

and$|f$“ $[\mathrm{Y}]_{\subset}^{2}|=1$.

In this note we are concerned with the case where $F$ is canonically defined, inparticular$\mathcal{F}_{\kappa\lambda}$, the minimal fine filter on$P_{\kappa}\lambda$.

We generally follow the terminology of Kanamori [Kan] with the following exception: For a cardinal$\mu\geq\omega$

we set $[X]^{\mu}=\{x\subset X : |x|=\mu\},$ $[X]^{<\mu}=\{x\subset X$ :1991 Mathematics Subject Classification.
.
Part of this work was done during the author’s stay at Boston University as one of the JapaneseOverseas Research Fellows. He gratefully acknowledges Prof. Kanamori’s hospitality. He also
wishes to thank Prof. Abe for helpful conversations.
$03\mathrm{E}05,03\mathrm{E}55$

Typeset by$A_{\mathcal{M}^{S-}\mathrm{E}\mathrm{X}}\mathrm{z}$23$|x|<\mu\}$

and $\lim A=\{\alpha<\mu : \sup(A\cap\alpha)=\alpha>0\}$ for

we mean$Xarrow(F^{+})_{2}^{2}$$a\cup b$of$a$for any $X\in F^{+}$ . We

and$\in[P_{\kappa}\lambda]_{\subset}m$Abe [A1] proved$b\in[P_{\kappa}\lambda]_{\subset}n$(see [JS]) to get the same conclusion
Proposition 1. Assume

Proof. First seton$\xi<\lambda$$P_{\kappa}\lambda=\{x_{\xi} :construct

andthattake$z_{\xi n}\in P_{\kappa}\lambda$$z_{\xi n}\cup\cup\{y_{\xi}^{\alpha i} :$f$defined byand$\mathrm{Y}_{\alpha}\in[X]^{\kappa}$$y_{\xi}^{\alpha i}\in \mathrm{Y}_{\alpha}$$P_{\kappa\lambda}\neq’(\mathcal{F}_{\kappa\lambda}^{+})_{2}^{2}$

and, andmutually distinct andfor$X\subset\{z_{\xi} :$\xi<\lambda$$\alpha\in z_{\xi n}$with\xi<\lambda\}$for any\alpha<\lambda\}$$\{z_{\xi} :$\xi<\lambda$. Thenwith$x_{\xi} \subset z_{\xi 0}\not\subset\bigcup_{\zeta<\xi}z_{\zeta}$$z_{\xi}= \bigcup_{n<\omega^{Z}\xi n}$$f$“$y_{\xi}^{\alpha i}\subseteq z_{\xi}$by induction on\xi<\lambda\}\neq’(\mathcal{F}_{\kappa\lambda}^{+})_{2}^{2}$. We show. By induction$y_{\xi}^{\alpha i}\in \mathrm{Y}_{\alpha}$

and $i<2$ so that

Finally set$\alpha\in z_{\xi}\in X$. We claim.

$[X]_{\subset}^{2}=2$. Take$f(\{y_{\xi}^{\alpha i}, Z\xi\})=i$$\alpha<\lambda$

for $i<2$ by$\square$

The above proof yields in fact for any“ $[\mathrm{Y}]_{\subset}^{2}=\gamma$.

$[P_{\kappa}\lambda]^{\kappa}=\{\mathrm{Y}_{\alpha} :witnesses$f(\{y_{\xi}^{\alpha i}, Z\xi\})=i$

definition, as desired.

$f$an opposite assumption:\alpha\in z_{\xi n} \mathrm{A} i<2\}_{\vee}\subset Z_{\xi 1}n+\cdot$

Fix an unbounded set
with\xi<\lambda\}$. Matet [Mat] used Laver’s idea

and $i<2$ as follows: At stage$\alpha\in z_{\xi}$$F^{+}arrow(F^{+})_{2}^{2}$

when the union$\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\cup a\subseteq\cap b$$\lambda^{<\hslash}=2^{\lambda}$$\mathrm{h}\mathrm{o}\mathrm{m}$. Then$x_{\xi}\subset z_{\xi}\in P_{\kappa}\lambda$mutually distinct for$n<\omega$$\lambda^{\kappa}=\lambda$. By

with $m,$ $n<\omega$ is formed.

under$\mathcal{F}_{\kappa\lambda}^{+}\neq’(\mathcal{F}_{\kappa\lambda}^{+})_{2}^{2}$$A\subset\mu$$\mathrm{Y}\in \mathcal{F}_{\kappa\lambda}^{+}$

with$\mathrm{Y}\subset X$

It is natural to ask, as did Abe [A1], if$\gamma<\kappa f$:$[X]_{\subset}^{2}arrow\gamma$

with$X\in \mathcal{F}_{\kappa\lambda}^{+}$

and.

$\mathcal{F}_{\kappa\lambda}^{+}\neq’(\mathcal{F}_{\kappa\lambda}^{+})_{2}^{2}$

holds in general. His answer[A2] to the analogous question would make it more interesting:$C_{\kappa\lambda}^{+}\star(C_{\hslash}^{+}\lambda)_{2}^{2}$.

Appealing more directly to Magidor’s idea [Mag], we give a canonical witness toAbe’s observation:
Proposition 2. Let

Proof. Set $S=$ {$\mu<\kappa$$x\in P_{\kappa}\lambda$

be regular. Then {$x\in \mathcal{P}_{\kappa}\lambda$: cf $(x\cap\kappa)=\mu$ }$\star(C_{\hslash}^{+}\lambda)_{2}^{2}$: cf $(x\cap\kappa)=\mu$} and for $x\in S$ fix an unbounded set.

$c_{x}\subset$

24$x\cap\kappa$of order type

$\mu$

. For

$\{x, y\}\in[S]_{\subset}^{2}$

and 1 otherwise. Fix a stationary set
First we have$\gamma\in c_{x}-C:y$$\gamma<\kappa$

andlet $f(\{X, y\})$ be$T\subset S$

such that for any$\gamma<\kappa$. We show$g(\gamma)\subset x,$ $y\iniff

T,$ $\gamma\in c_{x}$$\gamma\in c_{y}$. Take$f$when ni“ $[T]_{\subset}^{2}=2$$(c_{x}\Delta C_{y})\in c_{x}$,.

there are $w\subset x,$ $y\in T$ with

$w\in P_{\kappa}\lambda$

Suppose to the contrary that we have$0$

$g:\kappaarrow \mathcal{P}_{\kappa}\lambda$$x,$ $y\inC(g)\cap T$

such that for any
with$x\cap\kappa<y\cap\kappa$

by the stationarity of $\{z\cap\kappa:z\in C(g)\cap T\}$ . Then $c_{x}=c_{y}\cap X\cap\kappa$ has order type$\mu$

, contradicting the choice ofNow let$w_{\alpha}\subset x,$$w\subset x\subset y\subset z\mathrm{h}\mathrm{o}\mathrm{m}T$by.

be the minimal as above. Then for$\gamma<\kappa$

such that for any$c_{y}$$w_{\alpha}\subset x\subset y\subset z$, as desired.

$\mathrm{d}\dot{\mathrm{e}}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$y\in T,$

withfor any$\alpha\in c_{x}$iff$\alpha\in c_{y}$$\gamma\in c_{x}\cap c_{z}-c_{y}$$\alpha<\gamma$. Set. Then$\alpha<\gamma$

we have$w_{\alpha}\in P_{\kappa}\lambda$$w= \bigcup_{\alpha<\gamma}w_{\alpha}\in P_{\kappa}\lambda$. Take$\min(c_{xy}\Delta_{C})=\min(c_{yz}\Delta_{C})=\gamma$, and hence $f(\{x, y\})=0$ and $f(\{y, z\})=1$ by$\square$.The rest of the paper is devoted to a negative answer to Abe’s question in thestrong sense. We refer to Baumgartner’s expository paper [B] for the rudiments ofiterated forcings. We are indebted for the definition of the poset(see [JS]), who proved under$\mathrm{M}\mathrm{A}_{\omega_{1}}$that for any$f$:unbounded and $1<m,$ $n<\omega$ there is an unbounded

Assume for the moment that$f: \bigcup_{n<\omega}[S]^{n}\subsetarrow\gamma$

with$\kappa$$S\subset P_{\kappa}\lambda$$Q_{f}$$[X]_{\subset}^{m}arrow n$$\mathrm{Y}\subset X$

is a compact cardinal andbelow to Galvinwithwith$|f$$\lambda\leq 2^{\kappa}$$X\subset[\omega_{1}]^{<\omega}$“ $[\mathrm{Y}]^{m}\subset|=1$.

. Fix a coloring

unbounded and $1<\gamma<\kappa$ . We define a poset$Q_{f}$and establish its basic properties.

Fix a fine ultrafilter

on$[S]_{\subset}^{n}$

by

$U$

$U_{0}=\{\{\emptyset\}\}$For $n<\omega$ let$\beta_{n}$

on

$S$

and define inductively a -complete ultrafilter$\kappa$$U_{n}$

and $U_{n+1}=\{X : \{x : \{a : \{x\}\cup a\in X\}\in U_{n}\}\in U\}$ .

be the unique$\beta<\gamma$

with $\{a\in[S]_{\subset}^{n} : f(a)=\beta\}\in U_{n}$. Let25$Q_{f}--\{p\in[S]^{<\kappa} :$q\leq p$iff$q\supset p$\forall m,n<\omega\forall a\in[\mathrm{p}]_{\subset}m(\{b\in[S]_{\subset}^{n} :and$y\not\subset x$$\{\beta_{n}\}$\exists y\in q(x\subset y)\}$$G\subset Q_{f},$for any$\cup G$

is unbounded in$x\in \mathcal{P}_{\kappa}\lambda$

by the density of$P_{\kappa}\lambda$, and homogeneous for $f:f$ “$[\cup G]_{\subset}^{n}=$for any $n<\omega$ .

Next

is -centered closed (hence in particular -directed closed): A centered$Q_{f}$$\kappa$subset $D$ of$Q_{f}$$\kappa$of size$\alpha<\kappa$

has a lower bound $\cup D$ .

$<\kappa$

Finally we show thatwithandfor any $x\in p$ and $y\in q-p$ .

First for a generic filter$\{q\in Q_{f} :f(a\cup b)=\beta_{m+n}\}\in U_{n})\}$

is -linked. Fix an injection$Q_{f}$$\pi:P_{\kappa}\lambdaarrow\kappa_{2}$$\kappa$

set $Q_{f,A}=\{p\in Q_{f}$ : $\{\pi(x)|\alpha : x\in p\}=A$ A

is injective}. Then $Q_{f}=\cup\{Q_{f,A}$ :see linkedness of$Q_{fA},$, fix$\exists\alpha<\kappa(A\subset\alpha_{2)\}}$$p,$ $q\in Q_{f,A}$. Then$x\not\subset y$$\langle\pi(z)|\alpha :. Similarly$y\not\subset x$$A\subseteq\alpha_{2}$z\in\bigcup_{x\in p}PX\rangle$

by inaccessibility of . To$\kappa$

for any $x\in p-q$ and $y\in q$ :Otherwise we would have $x=z$ for some $x\in p-q,$ $y\in q$ with$\pi(x)|\alpha=\pi(Z)|\alpha$. For$x\subset y$

and $z\in q$ withfor any $x\in p$ and $y\in q-p$ . Thus $p\cup q\leq p,$ ,$q$

as desired.
A minor modification of the original proof [B] forLemma. Assume
that$\kappa^{+}$$\mathrm{I}\vdash_{\alpha}$“$\dot{Q}_{\alpha}$is$\kappa$$2^{<\kappa}=\kappa$. Let$\langle P_{\alpha},\dot{Q}_{\alpha} :\alpha<\beta\rangle$$\kappa=\omega_{1}$yields the folowing

be $a<\kappa$ -support iteration such

-linked and -centered closed” for any$\kappa$$\alpha<\beta$. Then$P_{\beta}$

satisfies

-c. .
$c$

Proof. Fix
any$\gamma<\kappa.$$\gamma_{\xi}^{p}<\kappa$”$\{\xi<\kappa :and$\delta<\kappa$$\alpha<\beta$

let$|\vdash_{\alpha}$

For $p\in X$ by induction onso thatandFor$X\in[P_{\beta}]^{\kappa^{+}}$$p_{\zeta}\leq p_{\xi}$\alpha_{\xi}^{p}=\alpha\}$so thatand“ $\dot{Q}_{\alpha}--\bigcup_{\gamma<\kappa}\dot{Q}_{\alpha}\gamma$$\xi<\kappa$take“$p_{\xi+1}|\alpha_{\xi}^{p}1\vdash_{\alpha_{\xi}^{\mathrm{p}}}$is unbounded for any$\delta\in\Delta_{\xi<\kappa}\cap\{\lim\{\xi<\kappa :$p_{\xi}\leq p,$$p_{\xi}(\alpha_{\xi})p\in\dot{Q}_{\alpha_{\xi}^{p}\gamma^{p}}\xi$”

withlinked for$\alpha_{\xi}^{p}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p_{\xi})$

andfor any $\xi<\zeta<\kappa$ ,. Take$\alpha\in\bigcup_{\xi<\kappa}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p\xi)$\alpha_{\xi}^{p}=\alpha\} :$\dot{Q}_{\alpha\gamma}$$\mathrm{Y}\in[X]^{\kappa^{+}}$\alpha\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p\epsilon)\}$for any26$p\in$

Y. Next takewith root$H=${$(\xi,$$d\in[\beta]^{<\kappa}$$\alpha_{\xi}^{p},$$\gamma^{p}\xi)$:so that$Z\in[\mathrm{Y}]^{\kappa^{+}}$. Fina.lly take$\xi<\delta$A$\alpha_{\xi}^{p}\in d$

To see that $W$ is linked, fixof$\{p_{\xi}, q_{\xi} :$\alpha\in s$

it suffices to show

When

$W\in[Z]^{\kappa^{+}}$$p,$ $q\inW$$\dot{Q}_{\alpha\gamma}$

and

$|f$

:

$1<\gamma<\kappa$, as desired. Otherwise the claim follows, since$r|\alpha|\vdash_{\alpha}$

is descending.”$\square$$F^{+}arrow(F^{+})_{\gamma}^{<\omega}$with $X\in F^{+}$ there isfor any $n<\omega$ . Note that$\mathrm{Y}\in F^{+}$$\mathcal{F}_{\kappa\kappa}^{+}arrow(F_{\kappa\kappa}^{+})_{\gamma}^{<}\omega$iff$\kappa$

we mean thatwith$\mathrm{Y}\subset X$

andis Ramsey for any.

Theorem. Let$\kappa^{+}$

forces$r|\alpha\leq p_{\xi+1}|\alpha,$ $q_{\xi+}1|\alpha$$\xi<\delta$$q_{\xi}(\alpha)\in$$\bigcup_{n<\omega}[x]_{\subset}^{n}arrow\gamma$“ $[\mathrm{Y}]_{\subset}^{n}|=1$$r$

is centered.”, for unboundedly many

We are now ready to prove our main result. By$f$so that$p_{\xi}(\alpha),$“ $\{p_{\xi}(\alpha),q_{\xi}(\alpha) : \xi<\delta\}\subset\dot{Q}_{\alpha}$

for any-system. At stage

by the choice of , and hence$(\xi, \alpha, \gamma)\in H$$\Delta$$H\in[\delta\cross d\cross\kappa]^{<\hslash}$“ $\{p_{\xi}(\alpha), q_{\xi}(\alpha) : \xi<\delta\}\subset\dot{Q}_{\alpha}$$r|\alpha|\vdash_{\alpha}$

forms a. Inductively we construct a lower bound$\alpha\in d=\bigcup_{\xi<\delta}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p\xi)\cap\bigcup_{\xi<\delta}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(q\xi)$, wherep\in Z\}$$s= \bigcup_{\xi<\delta}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}(p\xi)\cup\bigcup_{\xi<\delta}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(q\xi)$$\delta$$\alpha=\alpha_{\xi}^{\mathrm{p}}$\xi<\delta\} :} for any $p\in W$ .

with support\xi<\delta\}\subset P_{\beta}$$\{\{\alpha_{\xi}^{p} :$\kappa$

be a supercompact cardinal and-c. . poset forcing supercompactness of and$c$$\kappa$

a cardinal$\lambda$$>\kappa$$\mathcal{F}_{\kappa\lambda}^{+}arrow(\mathcal{F}_{\kappa\lambda}^{+})_{\gamma}^{<}\omega$. Then there is a

for any$1<\gamma<\kappa$.

Proof. First we force with the Laver poset [L] for and then add Cohen subsets$\lambda$$\kappa$of$\kappa$

to ensure supercompactness of

we perform a$<\kappa$$\kappa$-support iteration

for some canonical$P_{\alpha}$-nametogether with -closure and$\dot{f}$$P_{\alpha}$$\lambda\leq 2^{\kappa}$$\langle P_{\alpha},\dot{Q}_{\alpha} :in the further extensions. Next

\alpha<2^{\lambda^{<\kappa}}\rangle$

with$\mathrm{I}\vdash_{\alpha}$“ $\dot{Q}_{\alpha}=Q_{\dot{f}}$ ”

for a coloring. The standard inductive argument,$\kappa^{+}-\mathrm{c}.\mathrm{c}$$\kappa$the set of canonicaland. of$P_{\alpha}$

shows that-names for colorings for anyidentified with that of canonical$P_{2^{\lambda}}<\kappa$$P_{\alpha}$

is of size$\alpha<2^{\lambda^{<n}}$$\leq 2^{\lambda^{<\kappa}}$, and so is, whose union can be-names for colorings. Thus the iteration can

27

be arranged so that a homogeneous set for a coloring in the final model by$P_{2^{\lambda}}<\kappa$

appears in an intermediate model, which remains unbounded by absoluteness of$P_{\kappa}\lambda$, as desired.

$\square$

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$P_{\kappa}\lambda$$\kappa$$\kappa$$P_{\hslash}\lambda$

IN ST IT UTE OF M AT HEM AT IC S, UN IV ERS IT Y OF TSUKUBA, TsUKUBA, 305-8571 J A PA-mail addre8s:$E$$\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{o}\mathrm{y}\mathrm{a}\mathrm{m}_{\mathrm{a}}\mathrm{t}\mathrm{h}.\mathrm{t}$sukuba. ac. jp$\mathrm{N}$