If X is a set, [Χ]
ω
will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]
ω
, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]
ω
⊆ S or [Χ]
ω
∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.
The principal theorem of this paper is: Every Σ
1
1 (i.e., analytic) subset of [ω]
ω
is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ2
1 Π2
1 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ2
1 ∩ Π2
1, rather directly gives a Σ2
1 ∩ Π2
1 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ
2
1 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.