We study Ramsey’s theorem for pairs and two colours in the context of the theory of \(\alpha \)-large sets introduced by Ketonen and Solovay. We prove that any 2-colouring of pairs from an \(\omega ^{300n}\)-large set admits an \(\omega ^n\)-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama (Adv Math 330: 1034–1070, 2018) stating that Ramsey’s theorem for pairs and two colours is \(\forall \Sigma ^0_2\)-conservative over the axiomatic theory \({\textsf {RCA}}_{\textsf {0}}\) (recursive comprehension).

中文翻译：

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对对着色的有序值Ramsey数的一些上限

我们在Ketonen和Solovay引入的\（\ alpha \） -大集合理论的背景下研究对和两种颜色的Ramsey定理。我们证明\（\ omega ^ {300n} \） -大集合中的对的任何2色都允许\（\ omega ^ n \） -大的齐次集合。我们解释了此边界的形式化版本如何对Patey和Yokoyama的最新结果（Adv Math 330：1034-1070，2018）给出更直接的证明和加强，该结果指出对和两种颜色的拉姆西定理是\（ \ forall \ Sigma ^ 0_2 \） -关于公理理论的保守（\（{\ textsf {RCA}} _ {\ textsf {0}} \）（递归理解）。