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Ramsey monoids

Recently, Solecki [Forum Math. Sigma 7 (2019), p. 40] introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman’s theorem, Carlson’s theorem, and Gowers’ F I N k FIN_k theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey theorem. The results obtained suggest a possible connection with Schützenberger’s theorem and finite automata theory.

American Mathematical Society (AMS)

Claudio Agostini Eugenio Colla

Transactions of the American Mathematical Society

2023

Title: Ramsey monoids

Description:

Recently, Solecki [Forum Math.

Sigma 7 (2019), p.

40] introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman’s theorem, Carlson’s theorem, and Gowers’ F I N k FIN_k theorem.

He proved that an entire class of finite monoids is Ramsey.

Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids.

We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey theorem.

The results obtained suggest a possible connection with Schützenberger’s theorem and finite automata theory.

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