We study the effective and proof-theoretic content of the polarized Ramseyâ€™s theorem, a variant of Ramseyâ€™s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields aâ€¦ (More)

The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverseâ€¦ (More)

Weexamine the reversemathematics and computability theory of a formofRamseyâ€™s theorem in which the linear n-tuples of a binary tree are colored. Let 2<N denote the full binary tree of height!. Weâ€¦ (More)

Suppose thatf : [N]k â†’ N. A set A âŠ† N is free for f if for all x1, . . . , xk âˆˆ A with x1 < x2 < Â· Â· Â· < xk , f (x1, . . . , xk) âˆˆ A implies f (x1, . . . , xk) âˆˆ {x1, . . . , xk}. The free setâ€¦ (More)

We prove that if a Î 2 sentence is provable in a certain theory of higher order arithmetic without the law of the excluded middle then it is uniformly provable in the weak classical theory RCA0.â€¦ (More)

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concernâ€¦ (More)

Hirst, 1.L., Reverse mathematics and ordinal exponentiation, Annals of Pure and Applied Logic 66 (1994) 1-18. Simpson has claimed that "ATRo is the weakest set of axioms which permits the developmentâ€¦ (More)