Grant: Discovery Grants Program - Individual

Design theory and extremal set theory are the study of systems of sets (and other combinatorial objects) that satisfy some collection of properties (perhaps the sets must intersect in specific patterns or maybe the sets cannot be subsets of one another). This field of research typically includes finding constructions for specific types of systems and the determination of the size and structure of the maximum systems with a specific restriction. This is an important research area since these designs are often used in applications. For example, one design I study are covering arrays which are used to design effective test suites for software and networks. Because the current approaches to such problems are often ad hoc, a key theme of this research program is to develop a more systematic approach to these types of problems using tools from algebraic combinatorics and algebraic graph theory. My research has also brought into play elements from representation theory, group theory and computer algorithms.My current work is centered around what is probably the most famous theorem in extremal set theory, the Erdos-Ko-Rado theorem. This elegant theorem gives the size of the largest system of intersecting subsets of size k from a base set of n elements and describes exactly which systems meet this bound. There are many different proofs, generalizations, extensions and applications of this theorem. I am particularly interested in the generalizations of this theorem that apply to objects other than sets, and which have an algebraic proof.