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

TL;DR

This paper investigates the non-robustness of Ramsey's theorem and related principles in reverse mathematics, revealing how their strength varies with parameters like the number of colors and the structure of sets, through computability and reducibility analyses.

Contribution

It provides new insights into the non-robustness of Ramsey's theorem, including the impact of the number of colors and the separation of related combinatorial principles in reverse mathematics.

Findings

Ramsey's theorem is not robust with respect to the number of colors in computable reducibility.

Cohesiveness does not strongly reduce to stable Ramsey's theorem for pairs, highlighting combinatorial non-reducibility.

An infinite decreasing hierarchy of thin set theorems is identified, showing sensitivity of Ramsey's theorem's strength to the number of colors.

Abstract

Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study 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 stable Ramsey's theorem for pairs, revealing the combinatorial nature of this non-reducibility and prove that whenever $k$ is greater than $\ell$, stable Ramsey's theorem for $n$-tuples and $k$ colors is not computably reducible to Ramsey's theorem for $n$-tuples and $\ell$ colors. In this sense, Ramsey's theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey's theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.