Idempotents in compact semigroups and Ramsey theory
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.
scientific article; zbMATH DE number 4174988