A new and stronger central sets theorem

Dibyendu De¹ ; Neil Hindman² ; Dona Strauss³

¹ Department of Mathematics Krishnagar Women's College Krishnagar, Nadia-741101 West Bengal, India
² Department of Mathematics Howard University Washington, DC 20059, U.S.A.
³ Department of Pure Mathematics University of Leeds Leeds LS2 9J2, UK

Fundamenta Mathematicae, Tome 199 (2008) no. 2, pp. 155-175

Résumé

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.

Détail

DOI : 10.4064/fm199-2-5

Keywords: furstenbergs original central sets theorem applied central subsets mathbb finitely many specified sequences mathbb form already strong enough derive strong combinatorial consequences central subset mathbb contains solutions partition regular systems homogeneous equations subsequently central sets theorem extended apply arbitrary semigroups countably many specified sequences paper derive version central sets theorem arbitrary semigroups which applies sequences once version strictly stronger original version applied semigroup mathbb noncommutative versions strictly increasing strength