Effectiveness for the Dual Ramsey Theorem

TL;DR

This paper investigates the logical and computational strength of the Dual Ramsey Theorem for various colorings and partitions, establishing equivalences and bounds within reverse mathematics and effective analysis.

Contribution

It provides a detailed analysis of the Dual Ramsey Theorem's strength in reverse math, including equivalences for Baire and clopen colorings, and bounds on effective content for different cases.

Findings

Equivalence of Baire and clopen colorings for the theorem over RCA_0

Existence of computable Borel codes for colorings with high computational complexity

Partial results on bounds for the effective content of the theorem when k=2

Abstract

We analyze the Dual Ramsey Theorem for $k$ partitions and $\ell$ colors ($\mathsf{DRT}^k_\ell$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}_0$, the Dual Ramsey Theorem stated for Baire colorings is equivalent to the statement for clopen colorings and to a purely combinatorial theorem $\mathsf{cDRT}^k_\ell$. When the theorem is stated for Borel colorings and $k\geq 3$, the resulting principles are essentially relativizations of $\mathsf{cDRT}^k_\ell$. For each $\alpha$, there is a computable Borel code for a $\Delta^0_\alpha$ coloring such that any partition homogeneous for it computes $\emptyset^{(\alpha)}$ or $\emptyset^{(\alpha-1)}$ depending on whether $\alpha$ is infinite or finite. For $k=2$, we present partial results giving bounds on the effective content of the principle. A weaker version for $\Delta^0_n$ reduced colorings is equivalent to $\mathsf{D}^n_2$ over $\mathsf{RCA}_0+\mathsf{I}\Sigma^0_{n-1}$ and in the sense of strong Weihrauch reductions.