The Centrality of Central Sets
by
Neil Hindman
Howard University

Central sets in semigroups are guaranteed to have substantial combinatorial structure. For example, central subsets of ( N, +) must contain arbitrarily long arithmetic progressions and all of the finite sums of distinct terms of some infinite sequence, to mention two of the simplest structures that must be present.

I will argue that central sets should be interesting to topologists because topology is almost indispensable when dealing with them. Central sets were first defined by Furstenberg in terms of the dynamical notions of proximal , uniformly recurrent , and set of returns (not to mention the plain old topological notion of neighborhood ). Later, based on an idea of Bergelson, they were much more simply described in terms of minimal idempotents in the Stone-Cech compactification of a discrete semigroup. They also have a very complicated combinatorial characterization, but unlike other common notions of largeness in a discrete semigroup such as syndetic , thick , and piecewise syndetic they do not have a simple combinatorial characterization. So if one is working with central sets, one is essentially forced to use topology.

Date received: May 8, 2006