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 (mod k) if q is even, or, q ≡ 1 (mod 2 k) 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) , 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) , K 3 (G k (q)) . In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K 4 (G k (q)) (resp. K 3 (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 K 4 (G k (q)) and K 3 (G k (q)) , when q is prime, and Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]