# Generalized Paley Graphs, Finite Field Hypergeometric Functions and Modular Forms

Title: Generalized Paley Graphs, Finite Field Hypergeometric Functions and Modular Forms

Speaker: Dermot McCarthy (Texas Tech University)

Time: 10:30-11:30 Beijing time, Nov 22, 2022

Venue: BIMSA 1118

Zoom ID: 293 812 9202  Passcode: BIMSA

Abstract:

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, \emph{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).

BIMSA-YMSC Tsinghua Number Theory Seminar