Speaker: Mason Springfield (Texas Tech University)
Title: The number of transitive subtournaments of k-th power Paley digraphs and improved lower bounds for Ramsey numbers
Abstract: 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, 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. 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), 𝒦₄(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), 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 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 R₂(4) and R₃(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.