The number of 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 2k). We define the k-th power Paley digraph of order q, Gk(q), as the graph with vertex set Fq where a → b is an edge if and only if b − a is a k-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in Gk(q), K4(Gk(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 Gk(q), K3(Gk(q)). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K4(Gk(q)) (resp. K3(Gk(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 Rt(4), for all t ≥ 2, and for Rt(3), for all t ≥ 3.