Open questions about Ramsey-type statements in reverse mathematics

TL;DR

This paper discusses open questions in the combinatorics of Ramsey's theorem within reverse mathematics, highlighting gaps in understanding and the implications of unresolved issues for the field.

Contribution

It identifies and presents open questions about Ramsey-type statements, emphasizing gaps in current combinatorial and logical understanding.

Findings

Highlights unresolved questions in Ramsey's theorem combinatorics

Reveals gaps in reverse mathematics understanding

Suggests areas for future research

Abstract

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 extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. 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 inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.