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

Leszek Aleksander Kołodziejczyk ^[1] ; Keita Yokoyama ^[2]
1. [1] University of Warsaw
  
  University of Warsaw
  
  Warszawa, Polonia
2. [2] Japan Advanced Institute of Science and Technology
  
  Japan Advanced Institute of Science and Technology
  
  Japón
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 4, 2020
Idioma: inglés
DOI: 10.1007/s00029-020-00577-3
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Resumen
- 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).