Hindman's Coloring Theorem in arbitrary semigroups

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.