Pi04 conservation of Ramsey’s theorem for pairs
Published:
In this article, we prove that Ramsey’s theorem for pairs and two colors is a \( \forall \Pi^0_4\) conservative extension of \(\mathsf{RCA}_0 + \mathsf{B}\Sigma_2^0 \), where a \(\forall \Pi^0_4\) formula consists of a universal quantifier over sets followed by a \(\Pi^0_4 \) 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. For this, we introduce a new general technique for proving \(\Pi^0_4\)-conservation theorems.
Recommended citation: Q. Le Houérou, L. Levy Patey and K. Yokoyama (2024). "Pi04 conservation of Ramsey's theorem for pairs."
Download Paper