# Algebra Seminar talk

2023-03-17

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

Abstract:

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.