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

Abstract

In this article, we prove that Ramsey's theorem for pairs and two colors is $Π^1_1$-conservative over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \mathsf{WF}(ε_0)$ and over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \bigcup_n \mathsf{WF}(ω^ω_n)$. These results improve theorems from Chong, Slaman and Yang and Kołodziejczyk and Yokoyama and belong to a long line of research towards the characterization of the first-order part of Ramsey's theorem for pairs.