Abstract

. We study the proof--theoretic strength and e#ective 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 I#n and B#n denote the #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# 2 +RT 2 2 is conservative over RCA 0 +I# 2 for # 1 1 statements and that RCA 0 +I# 3 +RT 2 <# is # 1 1 -conservative over RCA 0 +I# 3 . It follows that RCA 0 +RT 2 2 does not imply B# 3 . In contrast, J. Hirst showed that RCA 0 +RT 2 <# does imply B# 3 , and we include a proof of a slightly strengthened version of this result. It follows tha...

Citations