Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

de Rancourt, N.

doi:10.1090/tran/8063

Ramsey theory without pigeonhole principle and the adversarial Ramsey principle
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Trans. Amer. Math. Soc. 373 (2020), 5025-5056 Request permission

Abstract:

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.

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