Ramsey Theory for Layered Semigroups

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.