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 α -large sets introduced by Ketonen and Solovay. We prove that any 2-colouring of pairs from an ω ^300n -large set admits an ω ^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 ∀Σ ^0_2 -conservative over the axiomatic theory _ (recursive comprehension).