Ramsey Theory: Central Sets and Related Combinatorially Rich Sets

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This research project concerns the mathematical subject of Ramsey theory. 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 specified sets (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 natural numbers N are finitely colored, there must be some infinite subset of N all of whose two-element subsets are the same color. Ramsey theory has applications in a wide variety of areas of mathematics as well as in theoretical computer science, communications, and information theory. The principal investigator will continue to mentor graduate students in projects on finite Ramsey Theory; this award consists solely of support for graduate students, facilitating the training through research involvement of the next generation of mathematicians.The principal investigator will continue his research in Ramsey theory, including the interactions between this part of combinatorics and the algebraic structure of the Stone-Cech compactification of a discrete semigroup. The principal investigator proved many years ago 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 is 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. 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 that satisfy the conclusion of the Central Sets Theorem are called C sets, and are much easier to describe in an elementary fashion. Investigation of these various algebraically characterized large subsets of N should continue to yield new Ramsey theoretic results.