The reverse mathematics of bounded Ramsey's theorem for pairs

Abstract

In this article, we study a degenerate version of Ramsey's theorem for pairs and two colors (${\mathsf{RT}}_2^2$), in which the homogeneous sets for color 1 are of bounded size. By ${\mathsf{RT}}_2^2$, it follows that every such coloring admits an infinite homogeneous set for color 0. This statement, called ${\mathsf{BRT}}_2^2$, is known to be computably true, that is, every computable instance admits a computable solution, but the known proofs use ${\Sigma }_2^0$-induction ($\mathsf{I}{\Sigma }_2^0$). We prove that ${\mathsf{BRT}}_2^2$ follows from the Erdős-Moser theorem but not from the Ascending Descending sequence principle, and that its computably true version is equivalent to $\mathsf{I}{\Sigma }_2^0$ over ${\mathsf{RCA}}_0$.

31 pages