Idempotents in Compact Semigroups and Ramsey Theory

We shall show here that van der Waerden’s theorem on arithmetic progressions and its variants, the Hales-Jewett theorem ([HJ]), the theorem of T. J. Carlson and S. G. Simpson ([CS]), and the theorem of T. J. Carlson (theorem 2 of [C]), can be obtained by an analysis of idempotents in compact semigroups. We obtain in this way a sharpening of Carlson’s result (which provides an “infinite” version of a theorem of Graham and Rothschild on k-parameter sets ([GR])) as well as an extension of Hindman’s theorem and the related theorem of K. Milliken and A. Taylor (cf. [H]). This extended result, which is the principal novel result of this paper, plays a crucial role in a density version of the Hales-Jewett theorem which we present elsewhere (see [FK]).