## Abstract

Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers a_{1}, a_{2}, . such that all of the sums ai1+ai2+aim (m≥1, i_{1}<i_{2}< <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}, . of S such that all but finitely many of the products ai1ai2 aim (m≥1, i_{1}<i_{2}< <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.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.

Original language | English |
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Pages (from-to) | 111-120 |

Number of pages | 10 |

Journal | Journal of Algebra |

Volume | 395 |

DOIs | |

State | Published - 1 Dec 2013 |

## Keywords

- Almost-monochromatic set
- Hindman Theorem in groups
- Hindman Theorem in semigroups
- Monochromatic semigroup
- Shevrin semigroup classification
- Synchronizing semigroup