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

Leszek Aleksander Kołodziejczyk, Keita Yokoyama

• 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 ω300n-large set admits an ωn-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 ∀Σ02-conservative over the axiomatic theory RCA0 (recursive comprehension).