Abstract
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).
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The work of the first author is partially supported by Grant No. 2017/27/B/ST1/01951 of the National Science Centre, Poland.
The work of the second author is partially supported by JSPS KAKENHI (Grant Nos. 16K17640 and 15H03634) and JSPS Core-to-Core Program (A. Advanced Research Networks), and JAIST Research Grant 2018 (Houga).
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Kołodziejczyk, L.A., Yokoyama, K. Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs. Sel. Math. New Ser. 26, 56 (2020). https://doi.org/10.1007/s00029-020-00577-3
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DOI: https://doi.org/10.1007/s00029-020-00577-3