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 $Σ^0_2$-induction ($\mathsf{I}Σ_2^0$). 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 $\mathsf{I}Σ_2^0$ over $\mathsf{RCA}_0$.