$\Pi^0_4$ conservation of Ramsey's theorem for pairs

Quentin Le Houérou; Ludovic Levy Patey; Keita Yokoyama

$\Pi^0_4$ conservation of Ramsey's theorem for pairs

Logic 2026-05-07 v3

Authors: Quentin Le Houérou , Ludovic Levy Patey , Keita Yokoyama

Abstract

In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall \Pi^0_4$ conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$ , 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.

Keywords

Ramsey's theorem $\Pi^0_4$ conservation reverse mathematics $\mathsf{RCA}_0$ $\mathsf{B}\Sigma^0_2$

Cite

@article{arxiv.2404.18974,
  title  = {$\Pi^0_4$ conservation of Ramsey's theorem for pairs},
  author = {Quentin Le Houérou and Ludovic Levy Patey and Keita Yokoyama},
  journal= {arXiv preprint arXiv:2404.18974},
  year   = {2026}
}

Comments

38 pages. Corrected the bound for largeness(RT_2^2) in Section 5.2

$\Pi^0_4$ conservation of Ramsey's theorem for pairs

Abstract

Keywords

Cite

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