Hindman's coloring theorem in arbitrary semigroups

Hindman's Theorem asserts that, for each finite coloring of $\N$, there are distinct $a_1, a_2, \dots \in \N$ such that all 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. Shevrin's classification of semigroups and a proof of Hindman's Theorem, due to Galvin and Glazer, imply together that, for each infinite semigroup $S$, there are distinct $a_1, a_2, \dots \in S$ 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 subsemigroup $T$ of $S$, such that all but finitely many members of $T$ have the same color.

Simple proofs. No background is needed.

Joint work with Boaz Tsaban.