Let k ≥ 2 be an integer. Let q be a prime power such that q ≡ 1 (mod k) if q is even, or, q ≡ 1 (mod 2k) if q is odd. The generalized Paley graph of order q, G k (q), is the graph with vertex set Fq where ab is an edge if and only if a − b is a k-th 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.

In honor of Lance Littlejohn on his 70th birthday.