Open questions about Ramsey-type statements in reverse mathematics

Ludovic Patey

Published 2015-06-15Version 1

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been studied under various reducibilities, namely, computable reducibility, uniform reducibility, reverse mathematics. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The unability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.