In 1955, Greenwood and Gleason proved that the two-color diagonal Ramsey number R(4,4) equals 18. Key to their proof was constructing a self-complementary graph of order 17 which does not contain a complete subgraph of order four. This graph is one in the family of graphs now known as Paley graphs. In the 1980s, Evans, Pulham and Sheehan provided a simple closed formula for the number of complete subgraphs of order four of Paley graphs of prime order.
Since then, generalized Paley graphs have been introduced. In this talk, we will discuss our recent work on extending the result of Evans, Pulham and Sheahan to generalized Paley graphs, using finite field hypergeometric functions. We also examine connections between our results and both multicolor diagonal Ramsey numbers and Fourier coefficients of modular forms.
This is joint work with Madeline Locus Dawsey (UT Tyler) and Mason Springfield (Texas Tech University).
Dermot McCarthy is an Associate Professor in the Department of Mathematics & Statistics at Texas Tech University. His main research interests are in number theory, with particular focus on modular forms and hypergeometric functions.