Large sets often have large partition regular structures-a final report
by
Neil Hindman
Howard University
Coauthors: Vitaly Bergelson (Ohio State University)

We show that for any of seven different notions of largeness in a semigroup S, there are numerous partition regular configurations for which, if B is large in S, then the set of such configurations which are contained in B is large in the same sense among the set of all such configurations. The notions of largeness have their origins in topological dynamics and the proofs use the algebra of the Stone-Cech compactification \betaS of S. When I gave earlier talks on this same subject, I was asked whether the proofs for these seven different notions of largeness could be unified. We show that for four of the notions, namely those which are duals of partition regular properties, the answer is ``yes''.

Date received: May 17, 1999