Hindman's coloring theorem in arbitrary semigroups

Gili Golan, Boaz Tsaban

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers a1, a2, . such that all of the sums ai1+ai2+aim (m≥1, i1<i2< <im) 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 a1, a2, . of S such that all but finitely many of the products ai1ai2 aim (m≥1, i1<i2< <im) 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 languageEnglish
Pages (from-to)111-120
Number of pages10
JournalJournal of Algebra
StatePublished - 1 Dec 2013


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


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