A new and stronger central sets theorem

Tom 199 / 2008

Dibyendu De, Neil Hindman, Dona Strauss Fundamenta Mathematicae 199 (2008), 155-175 MSC: Primary 05D10; Secondary 54H13, 22A15. DOI: 10.4064/fm199-2-5


Furstenberg's original Central Sets Theorem applied to central subsets of $\mathbb N$ and finitely many specified sequences in $\mathbb Z$. In this form it was already strong enough to derive some very strong combinatorial consequences, such as the fact that a central subset of $\mathbb N$ contains solutions to all partition regular systems of homogeneous equations. Subsequently the Central Sets Theorem was extended to apply to arbitrary semigroups and countably many specified sequences. In this paper we derive a new version of the Central Sets Theorem for arbitrary semigroups $S$ which applies to all sequences in $S$ at once. We show that the new version is strictly stronger than the original version applied to the semigroup $(\mathbb R,+)$. And we show that the noncommutative versions are strictly increasing in strength.


  • Dibyendu DeDepartment of Mathematics
    Krishnagar Women's College
    Krishnagar, Nadia-741101
    West Bengal, India
  • Neil HindmanDepartment of Mathematics
    Howard University
    Washington, DC 20059, U.S.A.
  • Dona StraussDepartment of Pure Mathematics
    University of Leeds
    Leeds LS2 9J2, UK

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