The weakness of being cohesive, thin or free in reverse mathematics

Ludovic Patey

Published 2015-02-12Version 1

A set is cohesive for a sequence of sets if it is almost contained in or in the complement of each set of the sequence. Cohesiveness plays a central role in the understanding of Ramsey-type principles in reverse mathematics. The cohesiveness principle is known to admit a universal instance, for which the cohesive sets have a jump of PA degree relative to 0'. In this paper, we investigate the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to the stable Ramsey's theorem for pairs, revealing the combinatorial nature of this non-reducibility and prove that whenever n is greater than m, the stable Ramsey's theorem for k-tuples and n colors is not computably reducible to the Ramsey's theorem for k-tuples and m colors. Finally, we separate the full free set theorem from Ramsey's theorem for pairs over omega-models, and answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.