Abstract. We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT n <∞ denote (∀k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X ≤ T 0 (n) . Let IΣ n and BΣ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + I Σ 2 + RT

Abstract. In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak König's Lemma).

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Abstract. We show that, over the base theory RCA 0 , Stable Ramsey's Theorem for Pairs implies neither Ramsey's Theorem for Pairs nor Σ 0 2 -induction.

Abstract. We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA 0 , and others are equivalent to ACA 0 . One, that every atomic theory has an atomic model, is not provable in RCA 0 but is incomparable with WKL 0 , more than Π 1 1 conservative over RCA 0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore (2007) that are not Π 1 1 conservative over RCA 0 . A priority argument with Shore blocking shows that it is also Π 1 1 -conservative over BΣ 2 . We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ 1 but implies IΣ 2 over BΣ 2 , and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ω-model consisting of the recursive sets.

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Abstract. One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[n] : n ∈ ω) approximating α and (b[n] : n ∈ ω) approximating β and a positive constant C such that for all n,

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