Logic Seminar
# Combinatorial statements from a compact right topological semigroup of types

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Europe/Rome

Aula Seminari (Dipartimento di Matematica)
### Aula Seminari

#### Dipartimento di Matematica

Description

Many theorems in combinatorics share a very similar structure: *Let $M$ be 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 a joint work with Eugenio Colla where we further extend the results obtained by Solecki.