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 ${\Pi }_1^1$-conservative over~${\mathsf{RCA}}_0+\mathsf{B}{\Sigma }_2^0+\mathsf{WF}({\epsilon }_0)$ and over~${\mathsf{RCA}}_0+\mathsf{B}{\Sigma }_2^0+{\bigcup }_n\mathsf{WF}({\omega }_n^{\omega })$. 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.

36 pages, fixed the proof of Lemma 3.25