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

In this article, we prove that Ramsey’s theorem for pairs and two colors is a $\mathrm{\forall }{\mathrm{\Pi }}_4^0$ conservative extension of ${\mathsf{R}\mathsf{C}\mathsf{A}}_0+\mathsf{B}{\mathrm{\Sigma }}_2^0$, where a $\mathrm{\forall }{\mathrm{\Pi }}_4^0$ formula consists of a universal quantifier over sets followed by a ${\mathrm{\Pi }}_4^0$ 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.