Hindman's Coloring Theorem in arbitrary semigroups

TL;DR

This paper extends Hindman's Coloring Theorem to arbitrary semigroups, characterizes semigroups with certain monochromatic subsemigroup properties, and connects these results to classical algebraic problems and Ramsey theory.

Contribution

It provides a characterization of semigroups where finite colorings admit infinite monochromatic subsemigroups, generalizing Hindman's Theorem beyond natural numbers.

Findings

Characterization of semigroups with monochromatic subsemigroups

Extension of Hindman's Theorem to arbitrary semigroups

Application of Ramsey's theorem to algebraic structures

Abstract

Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers $a_1,a_2,\dots$ such that all of the sums $a_{i_1}+a_{i_2}+\dots+a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. The celebrated Galvin--Glazer proof of Hindman's Theorem and a classification of semigroups due to Shevrin, imply together that, for each finite coloring of each infinite semigroup $S$, there are distinct elements $a_1,a_2,\dots$ of $S$ such that all but finitely many of the products $a_{i_1}a_{i_2}\cdots a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. Using these methods, we characterize the semigroups $S$ such that, for each finite coloring of $S$, there is an infinite \emph{subsemigroup} $T$ of $S$, such that all but finitely many members of $T$ have the same color. Our characterization connects our study to a classical problem of Milliken, Burnside groups and Tarski Monsters. We also present an application of Ramsey's graph-coloring theorem to Shevrin's theory.