Cohesive sets and rainbows

TL;DR

This paper investigates the logical strength of the Rainbow Ramsey Theorem for triples, establishing new recursion theoretic properties of cohesive sets and rainbows, and showing certain implications do not hold in base systems.

Contribution

It proves that $ ext{RCA} + ext{RRT}^3_2$ does not imply $ ext{WKL}$ or $ ext{RRT}^4_2$, and characterizes the Turing degrees of cohesive sets and rainbows for pair colorings.

Findings

Every sequence admits a cohesive set of non-PA Turing degree.

Every $ ext{0'}$-recursive sequence admits a $ ext{low}_3$ cohesive set.

Certain implications between principles do not hold in the base system.

Abstract

We study the strength of $\RRT^3_2$, Rainbow Ramsey Theorem for colorings of triples, and prove that $\RCA + \RRT^3_2$ implies neither $\WKL$ nor $\RRT^4_2$. To this end, we establish some recursion theoretic properties of cohesive sets and rainbows for colorings of pairs. We show that every sequence (2-bounded coloring of pairs) admits a cohesive set (infinite rainbow) of non-PA Turing degree; and that every $\emptyset'$-recursive sequence (2-bounded coloring of pairs) admits a $\low_3$ cohesive set (infinite rainbow).