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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

Author: N. de Rancourt
Journal: Trans. Amer. Math. Soc. 373 (2020), 5025-5056
MSC (2010): Primary 05D10, 03E60
Published electronically: April 28, 2020
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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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N. de Rancourt
Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot, Boîte Courrier 7012, 75205 Paris Cedex 13, France
Address at time of publication: Institut für Mathematik, Kurt Gödel Research Center, Universität Wien, Augasse 2-6, UZA 1 – Building 2, 1090 Wien, Austria

Keywords: Infinite-dimensional Ramsey theory, Gowers' games, determinacy
Received by editor(s): December 5, 2018
Received by editor(s) in revised form: November 1, 2019, and November 21, 2019
Published electronically: April 28, 2020
Additional Notes: This work was supported by EPSRC grant numbers EP/K032208/1 and EP/R014604/1.
Article copyright: © Copyright 2020 American Mathematical Society