A Partition Theorem for Perfect Sets

Let P be a perfect subset of the real line, and let the «-element subsets of P be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then P has a perfect subset whose «-element subsets he in at most (n — 1)! of the classes. Let C be the set of infinite sequences of zeros and ones, topologized as the product of countably many discrete two-point spaces, and ordered lexicographically. For X C C, let [X]n be the set of «-element subsets of X. When we describe a finite subset of C by listing its elements, we always assume that they are listed in increasing order. Thus, [C]" is identified with a subset of the product space C, from which it inherits a topology. A subset of C is perfect if it is nonempty and closed and has no isolated points. The purpose of this paper is to prove the following partition theorem, which was conjectured by F. Galvin who proved it [3] for n < 3. Theorem. Let P be a perfect subset of C and let [P]n be partitioned into a finite number of open (in [P]") pieces. Then there is a perfect set Q E P such that [Q]n intersects at most (n — 1)! of the pieces. Before setting up the machinery for the proof of this theorem, we point out some of its consequences. First, the therem remains true if C is replaced by the real line R with its usual topology and order. To see this, it suffices to observe that every perfect subset of R has a subset (a generalized Cantor set) homeomorphic to C via an order-preserving map and that any one-to-one continuous image in R of a perfect subset of C is perfect in R. Second, the hypothesis that the pieces of the partition are open can be greatly relaxed. Mycielski [6], [7] has shown that any meager set or any set of measure zero in [R]" is disjoint from [P]" for some perfect PçR. For the meager case, he obtains the same result with R replaced by any complete metric space X without isolated points. It follows that, if [R]" (or [X]") is partitioned into finitely many pieces that have the Baire property, then their intersections with [P]" are open in [P]" for some perfect P. Similarly, if the pieces are Lebesgue measurable, they become Gs sets when restricted to [P']n for suitable perfect 7"; since Gs sets have the Baire property, we can apply the preceding sentence, with 7" as X, to get a perfect PCP' such that the pieces intersected with [P]n are open in [P]". Thus, our theorem, as extended by the first remark above, implies the following partition theorem. Received by the editors April 27, 1979 and, in revised form, May 6, 1980; presented at a meeting on combinatorial set theory, Aachen, West Germany, June 1976. 1980 Mathematics Subject Classification. Primary 03E15, 03E05, 54H05. © 1981 American Mathematical Society 0002-9939/81/0000-0272/$02.75 271 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use