Conservation of Ramsey's theorem for pairs and well-foundedness

Quentin Le Houérou, Ludovic Levy Patey, Keita Yokoyama

Published 2024-02-18, updated 2024-08-29Version 2

In this article, we prove that Ramsey's theorem for pairs and two colors is $\Pi^1_1$-conservative over~$\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2 + \mathsf{WF}(\epsilon_0)$ and over~$\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2 + \bigcup_n \mathsf{WF}(\omega^\omega_n)$. These results improve theorems from Chong, Slaman and Yang and Ko{\l}odziejczyk and Yokoyama and belong to a long line of research towards the characterization of the first-order part of Ramsey's theorem for pairs.