Ramsey and Freeness Properties of Polish Planes

Let X be a Polish space, that is, a separable, completely metrizable topological space. We always assume that X is non-empty and perfect, that is, X has no isolated points. An old theorem of Mycielski (see [8]) says that given a sequence of meager sets …Bn Í X k…n†: n < q†, where 1 < k…n† < q, there exists a perfect set P Í X such that P is free for each Bn , that is, P k…n† Ç Bn Í f…x0 ; . . . ; xk…n†ÿ1†: xi ˆ xj for some i 6ˆ jg: Recall that P Í X is perfect if P is non-empty, closed and has no isolated points. Although the proof of this result is quite elementary, it has a number of important applications. Let us mention three of them. The ®rst one is an observation of Galvin (see [4, Theorem 19.6, p. 130]). It says that if E Í X , where 1 < n < q, has the Baire property and is not meager, then there exist perfect sets P0 ; . . . ;Pnÿ1 such that the perfect block Q i < n Pi is contained in E. Therefore, if X n ˆ S i < q Ai where the Ai have the Baire property (or even are Borel) and are pairwise disjoint (we shall refer to this situation as a Baire (Borel) colouring of X n by countably many colours), some Ai contains an n-dimensional perfect block (the block is monochromatic). Another theorem of Galvin (see [4, Theorem 19.7, p. 130]) says that if ‰X Š ˆ A0 È . . . È Akÿ1 is a ®nite partition of the two-element subsets of X such that each Ai has the Baire property in the sense that A i ˆ f…x; y† 2 X : fx; yg 2 Aig has the Baire property in X , then there exists a perfect set P Í X with ‰PŠ Í Ai for some i < k, that is, P is homogeneous for the partition. This result has been generalized to arbitrary ®nite dimension n by Galvin (for n ˆ 3) and by Blass (see [1]) (for arbitrary n). A notion of type for ntuples of reals is de®ned and the existence of a perfect set is proved such that any of its n-tuples of the same type have the same colour (for some ®xed Baire colouring of ‰X Š by ®nitely many colours). A third application of Mycielski's result (see [6, 2]) says that every Baire measurable function f : X n ! X has a free perfect set P Í X, which means that either f …x0 ; . . . ; xnÿ1† 2 fx0 ; . . . ; xnÿ1g or f …x0 ; . . . ; xnÿ1† 62 P, for all …x0 ; . . . ; xnÿ1† 2 P . In the present paper we consider spaces X which are not j-compact. Recall Hurewicz' Theorem (see [4, Theorem 7.10, p. 39]) which says that a Polish space is not j-compact if and only if it contains a homeomorphic copy of the Baire space q as a closed subspace. As perfect sets are typically compact, in this case