On the strength of Ramsey's theorem for pairs

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkn denote Ramsey's theorem for k -colorings of n-element sets, and let RT<<infinity>n denote(For Allk)RTkn. Our main result on computability is: For any n greater than or equal to 2 a nd any computable (recursive) k-coloring of the n-element sets of natural n umbers, there is an infinite homogeneous set X with X " less than or equal to (T) 0((n)). Let I Sigma (n) and B Sigma (n) denote the Sigma (n) inducti on 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 Sigma (2) + RT22 is conservative over RCA(0) + I Sigma (2) for Pi (1)(1) statements and that RCA(0) + I Sigma (3) + RT<<infinity>2 is Pi ( 1)(1)-conservative over RCA(0) + I Sigma (3). It follows that RCA(0) + RT22 does not imply B Sigma (3). In contrast, J. Hirst showed that RCA(0) + RT< <infinity>2 does imply B Sigma (3), and we include a proof of a slightly st rengthened version of this result. It follows that RT<<infinity>2 is strict ly stronger than RT22 over RCA(0).