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MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY

Part of: Extremal combinatorics Graph theory Semigroups

Published online by Cambridge University Press:  29 January 2019

SŁAWOMIR SOLECKI
SŁAWOMIR SOLECKI*
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, USA; ssolecki@cornell.edu

Abstract

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First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem.

MSC classification

Primary: 05D10: Ramsey theory 05C55: Generalized Ramsey theory 20M32: Algebraic monoids 20M99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e2
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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