Ramsey's theorem and recursion theory

Abstract

Let N be the set of natural numbers. If A ⊆ N , let [ A ] n denote the class of all n -element subsets of A . If P is a partition of [ N ] n into finitely many classes C 1 , …, C p , let H ( P ) denote the class of those infinite sets A ⊆ N such that [ A ] n ⊆ C i for some i . Ramsey's theorem [8, Theorem A] asserts that H ( P ) is nonempty for any such partition P . Our purpose here is to study what can be said about H ( P ) when P is recursive, i.e. each C i , is recursive under a suitable coding of [ N ] n . We show that if P is such a recursive partition of [ N ] n , then H ( P ) contains a set which is Π n 0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [ N ] n into two classes such that H ( P ) contains no Σ n 0 set. These results answer a question raised by Specker [12]. A basic partition is a partition of [ N ] 2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.