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In this article, we prove that Ramsey's theorem for pairs and two colors is a ∀ Π⁰₄ conservative extension of RCA₀ + BΣ⁰₂, where a ∀ Π⁰₄ formula consists of a universal quantifier over sets followed by a Π⁰₄ 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.
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Houérou et al. (2024) studied this question.
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