Effectiveness for the Dual Ramsey Theorem

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.