Title: An Algebraic Approach to Ramsey Theory

Abstract: Ramsey Theory has long been a field of combinatorics which has provided some particularly elusive problems. Erdős’s quip about the alien invasion gives some insight into just how difficult and computationally demanding these problems can be. In general, the Ramsey number R(r,s) is the smallest positive integer such that any red-blue edge-coloring of the complete graph on that many vertices has either a red K_r or a blue K_s. In this talk, we will discuss how this combinatorial problem can be translated into a question about a system of polynomial equations and explore the consequences to the Ramsey number of these systems.