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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 $\Sigma^0_2$-induction ($\mathsf{I}\Sigma_2^0$). We prove that $\mathsf{BRT}^2_2$ follows from the Erd\H{o}s-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 $\mathsf{RCA}_0$.