Generalized Paley graphs and their complete subgraphs of orders three and four

Abstract

Let $k \ge 2$ be an integer. Let q be a prime power such that $q \equiv 1 ({\mathrm{mod}}\,\,k)$ if q is even, or, $q \equiv 1 ({\mathrm{mod}}\,\,2k)$ if q is odd. The generalized Paley graph of order q, $G_k(q)$, is the graph with vertex set $\mathbb {F}_q$ where ab is an edge if and only if ${a-b}$ is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $G_k(q)$, $\mathcal {K}_4(G_k(q))$, which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original ($k=2$) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in $G_k(q)$, $\mathcal {K}_3(G_k(q))$. In both cases, we give explicit determinations of these formulae for small k. We show that zero values of $\mathcal {K}_4(G_k(q))$ (resp. $\mathcal {K}_3(G_k(q))$) yield lower bounds for the multicolor diagonal Ramsey numbers $R_k(4)=R(4,4,\ldots ,4)$ (resp. $R_k(3)$). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both $\mathcal {K}_4(G_k(q))$ and $\mathcal {K}_3(G_k(q))$, when q is prime, and Fourier coefficients of modular forms.