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' 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.Dedicated to the late Ronald L. Graham (1935Graham ( -2020. This paper would never have been written if not for his groundbreaking work in Ramsey theory.

Ramsey monoids

“…If M has a non-trivial class, then {0} ⊔ M is not almost R-trivial.The same procedure can be used more in general to "glue" R-rigid monoids in order to get new examples of R-rigid monoids. While almost R-trivial monoids have at most one non-trivial class, with this process one may create R-rigid monoids where all R-classes different from[1] R are non-trivial. Remark Let (S 1 , * 1 ), .…”