Separating principles below Ramsey's Theorem for Pairs

TL;DR

This paper investigates the logical strength of principles below Ramsey's Theorem for Pairs, clarifying their relationships and separating certain subsystems in reverse mathematics.

Contribution

It proves that specific principles, namely ADS and CAC, are not equivalent, and that EM is distinct from RT, advancing understanding of the structure below Ramsey's Theorem.

Findings

ADS is not equivalent to CAC

EM is not equivalent to RT

Clarifies the hierarchy of principles below RT

Abstract

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from $\RT$, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below $\RT$ has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that $\ADS$ is not equivalent to $\CAC$ and that $\EM$ is not equivalent to $\RT$.