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

Let k≥ 2 be an integer. Let q be a prime power such that q≡1(modk) if q is even, or, q≡1(mod2k) if q is odd. The generalized Paley graph of order q, G_k(q) , is the graph with vertex set 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) , K₄(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) , K₃(G_k(q)). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K₄(G_k(q)) (resp. K₃(G_k(q))) yield lower bounds for the multicolor diagonal Ramsey numbers R_k(4) = R(4 , 4 , … , 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 K₄(G_k(q)) and K₃(G_k(q)) , when q is prime, and Fourier coefficients of modular forms.