Borel sets and Ramsey's theorem

Abstract

Definition 1. For a set S and a cardinal κ,In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ω ⊆ S or else [M]ω ⊆ 2ω − S.Erdös and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ω ⋂ S1 = ∅ or [M]ω ∩ S2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.

A Note on Regular Methods of Summability and the Banach-Saks Property

“…This last inclusion is true since if q E C and $ is a monotone enumeration of q, then $ = kj for j < J. Hence, (A is the set of those p such that zJco a&, is a Cauchy sequence if kj is a monotone enumeration of p.) By a theorem of F. Galvin and K. Prikry [3] there is q E P(w) such that either (I) for every I r q, t E P(u) -f E A, or (II) for every t C q, r E P(w) =+ I E B. For the sequence {ej)iSs either (b) holds (in case (II)) or in case (I) we shall indicate how to pick a subsequence of it for which (a) holds.…”

The Covering Property Axiom, CPA

“…The notion has been formally defined by Darji [40], though it was already studied by Galvin and Prikry in [55]. Instead of using the original definition for this class, we will use its characterization due to Nowik [97] in which we consider P(ω) as a Polish space by identifying it with 2 ω via the characteristic function.…”

An effective proof that open sets are Ramsey

“…If P is a partition and X is an infinite subset of ω, then X lands in P if every infinite subset of X is in P, and X avoids P if no infinite subset of X is in P. A partition P is Ramsey if there is an infinite set X that either lands in P or avoids P. The theorems we are interested in are of the form "every well-behaved partition is Ramsey." A number of authors have shown independently that if P is open in the usual topology then it is Ramsey (see [4]), and the conclusion has been extended to Borel sets by Galvin and Prikry [4] and analytic sets by Silver [8,2].…”