Strong reductions and combinatorial principles

TL;DR

This paper investigates strong reducibilities between combinatorial principles in second-order arithmetic, providing new results on their non-reducibility and introducing techniques for controlling their computability properties.

Contribution

It answers open questions about uniform and strong computable reductions between key combinatorial principles, extending previous results and introducing new methods.

Findings

$ extsf{SRT}^2_2$ is not uniformly or strongly reducible to $ extsf{D}^2_{<inite}$

$ extsf{COH}$ is not uniformly reducible to $ extsf{D}^2_{<inite}$

$ extsf{COH}$ is not strongly reducible to $ extsf{D}^2_2$

Abstract

This paper is a contribution to the growing investigation of strong reducibilities between $\Pi^1_2$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch (to appear) about uniform and strong computable reductions between various combinatorial principles related to Ramsey's theorem for pairs. Among other results, we establish that the principle $\mathsf{SRT}^2_2$ is not uniformly or strongly computably reducible to $\mathsf{D}^2_{<\infty}$, that $\mathsf{COH}$ is not uniformly reducible to $\mathsf{D}^2_{<\infty}$, and that $\mathsf{COH}$ is not strongly reducible to $\mathsf{D}^2_2$. The latter also extends a prior result of Dzhafarov (2015). We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.