Dr. Emily Heath, Iowa State University

The Erdős-Gyárfás Problem on Generalized Ramsey Numbers A (p, q)-coloring of a graph G is an edge-coloring of G (not necessarily proper) in which each p-clique contains edges of at least q distinct colors. We are interested in the function f(n, p, q), first introduced by Erdős and Shelah, which is the minimum number of colors needed for a (p, q)-coloring of the complete graph on n vertices. In 1997, Erdős and Gyárfás initiated the systematic study of this function. Among other results, they gave upper and lower bounds on f(n, p, p) which are still the best known bounds for general p today. In this talk, I will give an overview of this problem and describe recent improvements on the probabilistic upper bound of Erdős and Gyárfás for several small cases of p. This is joint work with Alex Cameron.