Effectiveness for the Dual Ramsey Theorem

Abstract

We analyze the dual Ramsey theorem for k partitions and ℓ colors (DRTℓk) in the context of reverse math, effective analysis, and strong reductions. Over RCA0, the dual Ramsey theorem stated for Baire colorings Baire-DRTℓk is equivalent to the statement for clopen colorings ODRTℓk and to a purely combinatorial theorem CDRTℓk. When the theorem is stated for Borel colorings and k≥3, the resulting principles are essentially relativizations of CDRTℓk. For each α, there is a computable Borel code for a Δα0-coloring such that any partition homogeneous for it computes ∅(α) or ∅(α−1) depending on whether α is infinite or finite. For k=2, we present partial results giving bounds on the effective content of the principle. A weaker version for Δn0-reduced colorings is equivalent to D2n over RCA0+IΣn−10 and in the sense of strong Weihrauch reductions.