Carlson’s theorem on variable words and Gowers’ $\mathrm{FIN}_k$ theorem are generalizations of Hindman’s theorem that involve a monoid action on a semigroup. In short, they state that for any finite coloring of a semigroup there is an infinite monochromatic “span set”. They differ in the choice of the monoid. In 2019, Solecki introduced the notion of Ramsey monoid to isolate the common underlying structure of these theorems. Then, he proved a number of results, among which some necessary and sufficient conditions for a monoid to be Ramsey.

In this talk, I will present some results from two joint works with Eugenio Colla, where we obtain a purely algebraic characterization of Ramsey monoids, and give some new sufficient and necessary conditions for a monoid to be $\mathbb{YY}$-controllable --- a more technical notion that refines the concept of Ramsey monoid and allows to generalize other theorems in combinatorics, like the Furstenberg--Katznelson Ramsey Theorem.