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

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).