Algebra in Stone-Cech Compactifications and its Combinatorial Applications

Howard University, Washington DC

Investigators

Abstract

ABSTRACT Principal Investigator: Hindman, Neil Proposal Number: DMS - 0852512 Institution: Howard University Title: Algebra in Stone-Cech Compactifications and its Combinatorial Applications This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). A right topological semigroup is a set S which is a semigroup and a topological space and has the property that multiplication on the right by any fixed element of S is continuous. If S is a discrete semigroup, then its Stone-Cech compactification is, in a natural way, a compact Hausdorff right topological semigroup. As is true of any compact Hausdorff right topological semigroup, this Stone-Cech compactification has a smallest two sided ideal, and this ideal usually has elaborate algebraic structure. This project involves the study of the algebraic structure of the Stone-Cech compactification and its smallest ideal and investigation of the combinatorial applications of that structure, primarily to 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 (or "coloring") of specified sets. 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. The principal investigator gained some fame many years ago when he 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. But in 1975, Fred Galvin and Steven 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, that is elements such that p + p = p. Since then, numerous other applications of the algebraic structure of Stone-Cech compactifications to Ramsey Theory have been found.