Abstract
We further develop the theory of layered semigroups, as introduced by Farah, Hindman and McLeod, providing a general framework to prove Ramsey statements about such a semigroup $S$. By nonstandard and topological arguments, we show Ramsey statements on $S$ are implied by the existence of coherent sequences in $S$. This framework allows us to formalise and prove many results in Ramsey theory, including Gowers' $\mathrm{FIN}_k$ theorem, the Graham–Rothschild theorem, and Hindman's finite sums theorem. Other highlights include: a simple nonstandard proof of the Graham–Rothschild theorem for strong variable words; a nonstandard proof of Bergelson–Blass–Hindman's partition theorem for located variable words, using a result of Carlson, Hindman and Strauss; and a common generalisation of the latter result and Gowers' theorem, which can be proven in our framework.
Highlights
Ramsey theory mathematically studies to what extent regular configurations appear in disorder
We develop a general framework to prove partition theorems about a layered semigroup S, assuming only the existence of certain “coherent” sequences in S
Idempotent ultrafilters are key to most applications of infinitary methods in Ramsey theory—their existence follows from the Ellis–Numakura lemma
Summary
Ramsey theory mathematically studies to what extent regular configurations appear in disorder. We work in the setting of layered semigroups, but we consider a different, much broader class of morphisms, called regressive maps Working in this setting, we develop a general framework to prove partition theorems about a layered semigroup S, assuming only the existence of certain “coherent” sequences in S. We develop a general framework to prove partition theorems about a layered semigroup S, assuming only the existence of certain “coherent” sequences in S This framework allows a general way to formulate and prove many fundamental results of Ramsey theory. Natural examples of maps on layered semigroups are generally regressive, this notion distills the essential Ramsey-theoretic properties of such maps. An infinitary, multivariable generalisation of Bergelson, Blass and Hindman’s partition theorem on located variable words (§7) These theorems imply a variety of other Ramsey-type results, including Hindman’s finite unions theorem, the Hales–Jewett theorem, and van der Waerden’s theorem. We will use uppercase Latin letters A, B, . . . , S, T, . . . for sets and semigroups, lowercase Latin letters s, t, . . . for elements thereof, and lowercase Greek letters α, β, . . . for elements of nonstandard extensions ∗S of semigroups
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