${\Pi }_4^0$ conservation of Ramsey's theorem for pairs

Abstract

In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall {\Pi }_4^0$ conservative extension of ${\mathsf{RCA}}_0+\mathsf{B}{\Sigma }_2^0$, where a $\forall {\Pi }_4^0$ formula consists of a universal quantifier over sets followed by a ${\Pi }_4^0$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step towards the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs.

38 pages. Corrected the bound for largeness(RT_2^2) in Section 5.2