Ramsey Theory: Central sets and related combinatorially rich sets

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Ramsey Theory is that part of combinatorics that deals with the question of what sort of homogeneous structures one can expect to find in some one cell of a finite partition of a specified set (or sometimes in any suitably "large" subset). For example, the simplest nontrivial instance of the infinite version of Ramsey's Theorem says that whenever the two-element subsets of the set N of positive integers are finitely colored, there must be some infinite subset of N all of whose two element subsets are the same color. Many years ago, the principal investigator proved that whenever N is finitely colored, there must exist in one color an infinite sequence together with all of its finite sums of distinct terms without repetition. The original proof was elementary, but very complicated. Subsequently, other proofs were found that were less complicated. But in 1975, F. Galvin and S. Glazer showed that this "Finite Sums Theorem" is a completely trivial consequence of the fact that the Stone-Cech compactification of N can be given an algebraic structure extending ordinary addition which makes it a compact right topological semigroup, and therefore has idempotents. Sets with the property that they contain all the finite sums from a sequence are called IP sets. By virtue of the connection discovered above, a set is an IP set if and only if it has an idempotent in its closure in the Stone-Cech compactification of N. Those that have special idempotents which are called "minimal" in their closure are "central" sets. These sets have much stronger properties, many of which are consequences of the Central Sets Theorem. But central sets have a very complicated elementary description. Sets which satisfy the conclusion of the Central Sets Theorem are called "C-sets", and are much easier to describe in an elementary fashion. The proposed investigation of these various algebraically characterized large subsets of N should continue to yield new Ramsey-theoretic results. A significant portion of the funds in this grant will provide support for graduate students at Howard University, an historically black university. In particular, the grant will provide stipends for three Ph.D. students, two of whom are black Americans, both female. This project will therefore be instrumental in training mathematicians who come from a population that is severely underrepresented within the population of US mathematicians.