The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs

TL;DR

This paper investigates the logical strength of certain combinatorial principles related to Ramsey's Theorem for pairs, revealing implications and limitations within reverse mathematics and computability theory.

Contribution

It establishes new implications between principles like SRT$^2_2$, DNR, and COH, and answers an open question about stable colorings and homogeneous sets.

Findings

SRT$^2_2$ implies DNR over RCA$_0$

COH does not imply DNR over RCA$_0$

Every computable stable 2-coloring of pairs has an incomplete $ ext{Delta}^0_2$ infinite homogeneous set

Abstract

We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $\Delta^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.