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 C1, …, Cp, let H(P) denote the class of those infinite sets A ⊆ N such that [A]n ⊆ Ci 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 Ci, 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 Πn0 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 Σn0 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.

(Un)countable and (non)effective versions of Ramsey’s theorem

“…Jockusch [7] also showed that this cannot be avoided. His construction uses the following limit lemma twice.…”

“…But this theorem is highly non-constructive since there are computable infinite graphs none of its infinite cliques and anticliques is computable (they are not even in Σ 0 2 [7]). On the positive side, Jockusch also showed that every infinite computable graph contains an infinite clique or anticlique from Π 0 2 .…”

Ordinal Analysis and the Infinite Ramsey Theorem

“…It is known from work of Jockusch [5,Theorem 5.7] that iRT is not provable in ACA 0 . More precisely, for every n ≥ 0, there is a recursive F : [N] n+2 → 2 such that the n-th Turing jump of ∅ is recursive in any infinite F -monochromatic X ⊆ N. On the other hand, it also follows from [5,Theorem 5.6] that for every recursive F : [N] n → b and n ≥ 0 there exists an F -monochromatic X recursive in the n-th Turing jump of ∅.…”

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AbstractThe infinite Ramsey theorem is known to be equivalent to the statement 'for every set X and natural number n, the n-th Turing jump of X exists', over RCA0 due to results of Jockusch [5]. By subjecting the theory RCA0 augmented by the latter statement to an ordinal analysis, we give a direct proof of the fact that the infinite Ramsey theorem has proof-theoretic strength εω.

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Effectiveness for infinite variable words and the Dual Ramsey Theorem

“…Because the computable sets form the second order part of an ω-model of RCA 0 , the theorem implies that RCA 0 does not suffice to prove RT(2, 2). Jockusch [4] …”