Algebra Seminar talk

Claudio Agostini
From algebra to combinatorics through dynamics: some results in Ramsey theory

Many theorems in combinatorics share a very similar structure: Let $M$ be a monoid acting by endomorphism on a partial semigroup $S$. For each finite coloring of $S$, there are nice monochromatic subsets $N\subseteq S$. Examples of theorems of this form are Carlson’s theorem on variable words, Gowers’ $\mathrm{FIN}_k$ theorem, and Furstenberg-Katznelson's Ramsey theorem.

In 2019, Solecki isolated the common underlying structure of these theorems into a formal statement. Then, he proved several results, extending all aforementioned theorems at once. He also showed that such a statement strongly depends on the algebraic structure of the monoid and on the existence of certain idempotents in a suitable compact right topological semigroup.

In this talk, I will present joint work with Eugenio Colla where we further extend the results obtained by Solecki.