Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs

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 (2018), 1034--1070] stating that Ramsey's theorem for pairs and two colours is $\forall {\Sigma }_2^0$-conservative over the axiomatic theory ${\mathsf{RCA}}_0$ (recursive comprehension).

15 pages