Ramsey's theorem for singletons and strong computable reducibility

TL;DR

This paper proves that Ramsey's theorem for singletons with more colors is not strongly computably reducible to the stable Ramsey's theorem for fewer colors, showing a fundamental separation in computability strength.

Contribution

It establishes a strong non-reducibility result between Ramsey's theorem for singletons and the stable Ramsey's theorem, answering open questions in computability theory.

Findings

For any k > l, there exists a coloring c such that no stable coloring d computable from c yields an infinite homogeneous set computing one for c.

The cohesive principle is not strongly computably reducible to the stable Ramsey's theorem for all colorings.

This result provides the strongest known partial answer to whether the cohesive principle follows from the stable Ramsey's theorem in ω-models.

Abstract

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for $\ell$-colorings, $\mathsf{SRT}^2_\ell$. Our proof actually establishes the following considerably stronger fact: given $k > \ell$, there is a coloring $c : \omega \to k$ such that for every stable coloring $d : [\omega]^2 \to \ell$ (computable from $c$ or not), there is an infinite homogeneous set $H$ for $d$ that computes no infinite homogeneous set for $c$. This also answers a separate question of Dzhafarov, as it follows that the cohesive principle, $\mathsf{COH}$, is not strongly computably reducible to the stable Ramsey's theorem for all colorings, $\mathsf{SRT}^2_{<\infty}$. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether $\mathsf{COH}$ is implied by the stable Ramsey's theorem in $\omega$-models of $\mathsf{RCA}_0$.