We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkndenote Ramsey's theorem fork–colorings ofn–element sets, and let RT<∞ndenote (∀k)RTkn. Our main result on computability is: For anyn≥ 2 and any computable (recursive)k–coloring of then–element sets of natural numbers, there is an infinite homogeneous setXwithX″ ≤T0(n). LetIΣnandBΣndenote the Σninduction and bounding schemes, respectively. Adapting the casen= 2 of the above result (whereXis low2) to models of arithmetic enables us to show that RCA0+IΣ2+ (...) RT22is conservative over RCA0+IΣ2for Π11statements and that RCA0+IΣ3+ RT<∞2is Π11-conservative over RCA0+IΣ3. It follows that RCA0+ RT22does not implyBΣ3. In contrast, J. Hirst showed that RCA0+ RT<∞2does implyBΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2is strictly stronger than RT22overRCA0. (shrink)

We study lowness for genericity. We show that there exists no Turing degree which is low for 1-genericity and all of computably traceable degrees are low for weak 1-genericity.

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of (...) pairs admits a pair of low₂ infinite homogeneous sets whose degrees form a minimal pair. (shrink)

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey's Theorem for Pairs is not strong enough to prove , the comprehension scheme for arithmetical formulas, within the base theory , the comprehension scheme for recursive formulas. (...) We also show that Ramsey's Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within . In particular, Ramsey's Theorem for Pairs is not conservative over for -sentences. (shrink)