Separating Principles Below Ramseys Theo

In this paper, we investigate principles related to Ramsey’s Theorem for Pairs (RT2 2) within the context of reverse mathematics. The objective is to address two open questions regarding specific subsystems beneath RT2 2, namely demonstrating that ADS is not equivalent to CAC and that EM does not imply RT2 2. To achieve this, we commence with a systematic review of the known results and definitions pertaining to relevant subsystems, while operating under the base theory RCA0. Our methodology involves constructing examples and counterexamples to illustrate the distinctions and implications of these principles. The results reveal a complex landscape of relationships between the subsystems, affirming that ADS cannot be derived from CAC, and EM is insufficient for deriving SRT2 2. Consequently, our findings contribute significantly to the understanding of the hierarchy of principles in reverse mathematics and the nuances between these critical systems. We further establish a Turing ideal that satisfies EM yet fails to satisfy SRT2 2, thereby reinforcing our primary conclusions.