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Actions of trees on semigroups, and an infinitary Gowers-Hales-Jewett Ramsey theorem

Lupini, Martino (2019) Actions of trees on semigroups, and an infinitary Gowers-Hales-Jewett Ramsey theorem. Transactions of the American Mathematical Society, 371 (5). pp. 3083-3116. ISSN 0002-9947. doi:10.1090/tran/7337.

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We introduce the notion of (Ramsey) action on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales-Jewett theorems (for both located and nonlocated words), and the Farah-Hindman-McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken-Taylor theorem of Bergelson-Hindman-Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of recurrence sets in amenable groups.

Item Type:Article
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Lupini, Martino0000-0003-1588-7057
Additional Information:© 2018 American Mathematical Society. Received by the editors November 27, 2016, and, in revised form, June 12, 2017 and July 19, 2017. Article electronically published on December 3, 2018. The author was supported by the NSF grant DMS-1600186. We would like to thank Andy Zucker and Joel Moreira for their comments on a preliminary version of the present paper. We are also grateful to the anonymous referee for carefully reading the paper and providing many useful remarks and suggestions.
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Subject Keywords:Gowers Ramsey theorem, Galvin–Glazer–Hindman theorem, Milliken–Taylor theorem, Hales–Jewett theorem, tetris operation, variable word, partial semigroup, ultrafilter, Stone–Čech compactification.
Issue or Number:5
Classification Code:2010 Mathematics Subject Classification. Primary 05D10, 54D80; Secondary 20M99, 05C05, 06A06
Record Number:CaltechAUTHORS:20180410-161934797
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85739
Deposited By: Tony Diaz
Deposited On:11 Apr 2018 14:58
Last Modified:15 Nov 2021 20:31

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