Borel Combinatorics Seminar

Claudio Agostini (TU Wien)

Title: On Ramsey monoids

Time: Thursday, 16.05.2024, 16:15.
Place: ELTE, D3-306.
Abstract: 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 $N\subseteq S$''. Notable examples include 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, and then, he proved a number of results, showing that the possibility to obtain such a statement 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 introduce these different notions and explain the known relations between them and with other algebraic classes of monoids.