Strong Reductions between Relatives of the Stable Ramsey's Theorem

TL;DR

This paper thoroughly analyzes the computable reductions among various variants of the Stable Ramsey's Theorem, establishing precise relationships and non-reducibility results between them.

Contribution

It provides a complete characterization of the reducibility relations between different stable Ramsey-type principles.

Findings

Established strict non-reducibility between certain principles.

Mapped the reducibility hierarchy among the variants.

Clarified the computational strength differences between the principles.

Abstract

A complete analysis is given of the computable reductions that hold between $\mathsf{SRT}^2_2$, $\mathsf{SPT}^2_2$, and $\mathsf{SIPT}^2_2$. In particular, while $\mathsf{D}^2_2\le_{\rm sW}\mathsf{SIPT}^2_2\le_{\rm sW}\mathsf{SPT}^2_2\le_{\rm sW}\mathsf{SRT}^2_2$, it is shown that $\mathsf{SRT}^2_2\not\le_{\rm sc}\mathsf{SPT}^2_2\not\le_{\rm sc}\mathsf{SIPT}^2_2\not\le_{\rm sc}\mathsf{D}^2_2$.