We analyze the dual Ramsey theorem for k partitions and ℓ colors () in the context of reverse math, effective analysis, and strong reductions. Over , the dual Ramsey theorem stated for Baire colorings is equivalent to the statement for clopen colorings and to a purely combinatorial theorem . When the theorem is stated for Borel colorings and , the resulting principles are essentially relativizations of . For each α, there is a computable Borel code for a -coloring such that any partition homogeneous for it computes or depending on whether α is infinite or finite. For , we present partial results giving bounds on the effective content of the principle. A weaker version for -reduced colorings is equivalent to over and in the sense of strong Weihrauch reductions.
"Effectiveness for the Dual Ramsey Theorem." Notre Dame J. Formal Logic 62 (3) 455 - 490, August 2021. https://doi.org/10.1215/00294527-2021-0024