Idempotents in compact semigroups and Ramsey theory
type: Item
instance of: scholarly article
title: Idempotents in compact semigroups and Ramsey theory
zbMATH Open document ID: 0714.05059
DOI: 10.1007/BF02764984
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: In this paper, by analysing idempotents in compact semigroups, the authors give a new proof of van der Waerden's theorem on arithmetic progressions as well as its variants, the Hales-Jewett theorem [\textit{A. W. Hales} and \textit{A. I. Jewett}, ``Regularity and positional games.'' Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)], the theorem of Carlson and Simpson [\textit{T. J. Carlson} and \textit{S. G. Simpson}, ``A dual form of Ramsey's theorem.'' Adv. Math. 53, 265--290 (1984; Zbl 0564.05005)], and the theorem of Carlson [theorem 2 of \textit{T. J. Carlson}, ``Some unifying principles in Ramsey theory.'' Discrete Math. 68, No. 2--3, 117--169 (1988; Zbl 0817.04002)]. Moreover, they obtain in this way a sharpening of Carlson's result as well as an extension of Hindman's theorem and the related theorem of K. Milliken and A. Taylor [cf. \textit{N. Hindman}, ``Ultrafilters and combinatorial number theory.'' In: Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 119--184 (1979; Zbl 0416.10042)]. They also treat the infinitary versions of the Hales-Jewett theorem.
reviewed by: Jiahai Kan
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 05C55
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 4174988
zbMATH Keywords: idempotents
zbMATH Keywords: compact semigroups
zbMATH Keywords: Ramsey theory
zbMATH Keywords: Hindman's theorem
author: Hillel Furstenberg
author: Yitzhak Katznelson
MaRDI profile type: Publication
cites work: Nonmetrizable Topological Dynamics and Ramsey Theory
cites work: Some unifying principles in Ramsey theory
cites work: A dual form of Ramsey's theorem
cites work: Q5589084
cites work: A density version of the Hales-Jewett theorem for \(k=3\)
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cites work: Regularity and Positional Games
cites work: Q3205333
full work available at URL: https://doi.org/10.1007/bf02764984
OpenAlex ID: W2003723973
description: scientific article; zbMATH DE number 4174988