Pi04 conservation of Ramsey’s theorem for pairs
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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: Le Houérou, Q., Levy Patey, L. and Yokoyama, K. (2026). "Pi04 conservation of Ramsey's theorem for pairs." J. London Math. Soc., 113: e70419. https://doi.org/10.1112/jlms.70419
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