Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

Noé de Rancourt (1, 2, 3, 4, 5)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 UK - Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic]
3 Universität Wien = University of Vienna
4 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
5 DMA - Département de Mathématiques et Applications - ENS Paris

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.

Mots clés: Infinite-dimensional Ramsey theory; Gowers' games; Determinacy

Domaines: Logique [math.LO]; Combinatoire [math.CO]

Noé de Rancourt. Ramsey theory without pigeonhole principle and the adversarial Ramsey principle. Transactions of the American Mathematical Society, 2020, 373 (7), pp.5025-5056. ⟨10.1090/tran/8063⟩. ⟨hal-04406796⟩