数理解析研究所講究録
第 1595 巻 2008 年 1-5

1Partitions of the reals and models of $ZF$
Carlos Augusto Di Prisco
Instituto Venezolano de Investigaciones Cient\’ificasFebruary 29, 2008
Abstract
We consider several partition relations and describe models of $ZF$
which can be used to distinguish between them. This is an extended
abstract of a talk delivered in the RIMS Symposium on Axiomatic Set
Theory and Set Theoretic Topology, held at RIMS University of Kyoto,
28-30 November 2008.

1Introduction.

of all infinite sequences of naturalWe consider partitions of the Baire spacenumbers with the product topology obtained giving to the discrete topolgy,of all infinite subsets of , which
and also partitions of its closed subspacecan be identified with the strictly increasing sequences of natural numbers. Ifto denote the set of infinite
is an infinite set of natural numbers, we use
subsets of .
$\omega^{w}$$\omega$$[\omega]^{w}$$\omega$$[A]^{w}$

$A$

$A$

, we say that a partition $c:[w]^{w}arrow n$ is Ramsey ifDefinition 1 Given. Such a set $H$ is said to besuch that is constant on
there is$n\in\omega$$H\in[\omega]^{w}$$homogeneou8$$[H]^{w}$$c$

for .
$c$

One of the emblematic results in this area is the following theorem of F.
Galvin and K. PrikryTheorem 2 $/5J$ For everyis Ramsey.

$n\in\omega$, every Borel measurable $parti\hslash onc:[w]^{w}arrow n$

The notation$\omegaarrow(w)_{n}^{w}r$is used to express that for every -measurable $c:[\omega]^{w}arrow n$ , there is. So, the Galvin-Prikry theorem issuch that is constant on$\Gamma$$c$$[H]^{\omega}$

物 $(\omegaarrow(\omega)_{n}^{w})$

Borel.

$H\in[\omega]^{w}$

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If no class is mentioned, the partition symbol refers to all functions. Also, if $n=2$ , the subindex is usually omitted.
It is well known that
implies that there are no non-principal ultrafilters on ; so, ZFC proves that this partition relation is false. Nevertheless,a celebrated result of Mathias [7] shows that this partition relation is consistent with $ZF+DC$ , provided that the existence of an inaccessible cardinal isconsistent.
$\Gamma$$c:[\omega]^{\omega}arrow$$n$$\omegaarrow(\omega)^{w}$$w$

2Infinite partitions.

It is easy to find a clopen non-Ramsey partition of
into infinitely manypieces. Namely, :defined by $h(A)= \min(A)$ . Thus, $ZF$ proves$[w]^{w}$$h$$[w]^{\omega}arrow w$$w\star(\omega)_{w}^{w}$

It is interesting to consider a version ofthat requires only theexistence of a set of the form
which avoids a piece of the partition, insteadof requiring that it is contained in a single piece. For this type of partition
relation it is customary to use the following notation. The expression$\omegaarrow(\omega)_{w}^{w}$$[H]^{w}$$warrow[w]_{K}^{\omega}r$means that for every -measurable $c:[w]^{w}arrow K$ , there is$\Gamma$$c’[H]^{w}\subsetneq K$.

$H\in[\omega]^{\omega}$such that

It is straightforward to $veri\Psi$ that this partition relation holds for Borel
partitions, but again, the Axiom of Choice implies that there are partitions ofinto infinitely many pieces for which every set of the form
meets every
piece. In fact, we have the following.
$[w]^{\omega}$$[H]^{w}$Proposition 3 If there is a $non- p\dot{n}ncipal$ ultrafilter on$\omega\neq\nu[\omega]_{2}^{w}.$$\omega$, then.

Actually, a weaker hypothesis is enough to refute the partition relation$warrow[w]_{2^{w}}^{w}$,.

namely, the existence of a non-principal non-meager filter onThis result ispart of ongoing work done jointly with S. Todorcevic and will appear elsewhere.
$w$3Homogeneous sublattices and perfect sets.

We now turn to a different type of partition property, which was first consideredin [4].
We use the symbol$warrow((\omega))_{n}^{\omega}\Gamma$

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to express that for every F-measurable function ;$A,$ $B\in[w]^{w}$ , wtih
, such that
and$c$$A\subseteq B$$B\backslash A\in[\omega]^{w}$

sublattice of subsets of given by
It is easily seen that the relation$B$$[\omega]^{w}arrow n$$c$, there areis constant on the$[A, B]=\{X\subseteq B : A\subseteq X\}$.

$warrow((\omega))_{n}^{w}Borel$

follows from$\omegarightarrow(w)_{n}^{w}$Borel.

And just as in the case of $warrow(w)^{w}$ , the existenoe of a non-principal ultrafilter on implies that $w\wedge((w))^{w}$ .
The third type of partition relation we consider here is denoted by$w$$warrow(perfect)_{n}^{w}r$meaning that for every -measurable function $c:[\omega]^{w}arrow n$ , there is aperfect
set
on whii is constant.
ABernstein set is just acounterexample to $warrow(perfect)^{w}$ , this is, aset $B$
with the property that both $B$ and its complement meet every perfect set. Suchaset can be obtained&om awell ordering of the reak.
$\Gamma$$P\subseteq[w]^{w}$$c$

In his article [8] Solovay, assuming the $consi_{8}tency$ of inaccesible cardinak,
constructed amodel of $ZF$ where every set of reals is Lebesgue measurable,
has the property of Baire, and if not countable, contains aperfect subset. Of
course, the axiom ofdoes not hold in this model, although the axiom of
dependent choices does. In general, amodel $M$ of ZF is said to be aSolovay
computed in the Lvy
model if it is (elementary equivalent to) the model
collapse of an inaccessible cardinal to. The result of Mathias mentioned$\bm{i}o\ddagger ce$$L(\mathbb{R})$$\aleph_{1}$above ([7])) establishes that the partition property $warrow(w)^{\omega}$ holds in Solovay,
models; therefore the same is true for the properties$\omegaarrow(perfect)^{w}$
$hom$
and
which follow
it.
Consider now the modelobtained adding aselective ultrffilter tousing the poset of infinite subsets of ordered by theaSolovay model
relation of almost containment.
. This was done
It was vhown in [2] that $\omegaarrow(perfect)^{\omega}$ holds inproving that in Solovay modek, the fofowing parameterized partition relation
holds: for every $n\in\omega$ and every $c:[w]^{w}xw^{w}arrow n$ , there is $H\in[w]^{w}\bm{t}d$ a.
perfect set
such that is constant on the productTherefore, the existence of anon-principal ultrafilter on is aconsequenceof the Axiom of Choice not strong enough to produce aBemstein set. By ourprevious remarks about non-principal ultrafilters, none of the other $propertie8$, since in it is non-principal ultraffiter on .
hold in the model$\omegaarrow((\omega))^{w},$$\omegaarrow[w]_{2^{\omega}}^{w}$$L(\mathbb{R})[\mathcal{U}]$$L(\mathbb{R})$$\omega$$L(\mathbb{R})[\mathcal{U}]$$P\subseteq w^{w}$$[H]^{\omega}\cross P$

$c$

$w$

$L(\mathbb{R})[\mathcal{U}]$$\mathcal{U}$

$w$

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Cohen extensions

Adding Cohen generic reals to the constructuble universe
in which$L$, we obtain a model$\omegaarrow((\omega))^{w}p_{rojective}$counterexample for.
holds but there is aproductWe start $homL$ , and add -many Cohen genric reak using theof Cohen forcing with finite support. In, [1] it is shown that in this extension
the partition relation $\omegaarrow((w))^{w}$ holds for projective partitions.
counterexample for
It follows $hom[6],$ $2.2$ , that in this model there is a$warrow(w)^{w}$ , i.e. there
non-Ramsey set.
afact,
hol&in the extension for partitions definable
the relation
Inso, it also holds in the inner modelof a\"u thewith real parameters,
sets in the extension that are constructible $hom$ reals. In this way we obtain adoes not.
model in which $warrow((\omega))^{w}$ holds but
The model obtained adding -many Cohen generic reak to offers additional features. For example, in this model there is anon-meager non-principal
filter on . Taking the appropriate inner model we obtain amodel in which$\omegaarrow((w))^{w}$ holds, but
faik.
$warrow(\omega)^{\omega}$$\Delta_{2}^{1}$$\omega_{1}$$\omega_{1}$$\Delta_{2}^{1}$$i_{8}$$\Delta_{2}^{1}$$\omegaarrow((\omega))^{\omega}$$\bm{t}d$$L(\mathbb{R})$$warrow(\omega)^{\omega}$$L$$\omega_{2}$$w$$\omegaarrow[w]_{2^{w}}^{w}$

5Conclusion.

,
implies bothandSumarizing, we have thatthe first implication being strict.
imply $warrow(perfect)^{w}$ ,Each of the properties $warrow((w))_{2}^{w}$ andis not impliedand both implications are strict. The partition relation
by $warrow((w))^{w}$ .
$warrow(\omega)^{w}$$\omegaarrow((\omega))^{w}$$\omegaarrow[w]_{2^{w}}^{\omega}$$\omegaarrow[\omega]_{2^{w}}^{w}$$warrow[w]_{2^{w}}^{w}$

Question: What is the exact relationvhip between the propereties $warrow(w)^{w}$
and $warrow[w]_{2}^{w}.$ ? (See [3]).

References[1] Brendle, J., L. Halbeisen and B. L\"owe, Silver measurability and and its
relation to other regularity properties. Math. Proc. Cambridge Philos.
Soc. 138 (2005), no. 1, 135-149[2] Di Prisco, C.A., Partition properties and perfect sets. Notas de L\’ogzca
Matem\’atica (Proceedings of the IX Latin American Symposium onMathematical Logic) 38 INMABB-CONICET. Universidad Nacionaldel Sur, Bah\’ia Blanca, Argentina (1993), 119-127.

5[3] Di Prisco, C. A. and J. Henle, Partitions of the reals and choice. in:Models, Algebras and Proofs (Xavier Caicedo and Carlos H. Montenegro, Eds.) Lecture Notes in Pure and Applied Mathematics Vol. 203
Marcel Dekker (1999) pp 13-23.

[4] Di Prisco, C. A. and J. Henle, Doughnuts, floating ordinals, square
brackets and ultraflitters. Joumal of Symbolic Logic 65 (2000) 461-473.

[5] Galvin, F. and K. Prikry, Borel sets and Ramsey’s theorem, Joumalof Symbolic Logic 38 (1973), 193-198.

[6] Ihoda, J.

642.

$I_{)}\Sigma_{2}^{1}$

sets of reals. Joumal of Symbolic Logic 53 (1988) 636-

[7] Mathias, A.R.D., Happy families. Annals of Pure and Applied Logic
12 (1977) 59-111.

[8] Solovay, R., A model of set theory in which every set of reals isLebesgue measurable. Annals of Mathematics 92 (1970) 1-56.