Transitive Subtournaments of k-th Power Paley Digraphs and Improved Lower Bounds for Ramsey Numbers

Let k ≥ 2 be an even integer. Let q be a prime power such that q ≡ k + 1 ( mod 2 k ) . We define the k-th power Paley digraph of order q , G k ( q ) , as the graph with vertex set F q where a → b is an edge if and only if b - a is a k -th power residue. This generalizes the ( k = 2 ) Paley Tournament. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in G k ( q ) , K 4 ( G k ( q ) ) , which holds for all k . We also provide a formula, in terms of Jacobi sums, for the number of transitive subtournaments 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 directed Ramsey numbers R k 2 ( 4 ) = R ( 4 , 4 , … , 4 ) (resp. R k 2 ( 3 ) ). We state explicitly these lower bounds for k ≤ 10 and compare to known bounds, showing improvement for R 2 ( 4 ) and R 3 ( 3 ) . Combining with known multiplicative relations we give improved lower bounds for R t ( 4 ) , for all t ≥ 2 , and for R t ( 3 ) , for all t ≥ 3 .