Effectiveness for the Dual Ramsey Theorem

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 RCA(0), 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 alpha, there is a computable Borel code for Delta(0)(alpha)-coloring such that any partition homogeneous for it computes theta((alpha)) or theta((alpha-1)) depending on whether ff 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 D-2(n) over RCA(0) + I Sigma(0)(n-1) and in the sense of strong Weihrauch reductions.