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

Abstract

Let \(k \ge 2\) be an even integer. Let q be a prime power such that \(q \equiv k+1 (\text {mod}\,\,2k)\). We define the k-th power Paley digraph of order q, \(G_k(q)\), as the graph with vertex set \(\mathbb {F}_q\) where \(a \rightarrow b\) 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 \(G_k(q)\), \(\mathcal {K}_4(G_k(q))\), 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 \(G_k(q)\), \(\mathcal {K}_3(G_k(q))\). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of \(\mathcal {K}_4(G_k(q))\) (resp. \(\mathcal {K}_3(G_k(q))\)) yield lower bounds for the multicolor directed Ramsey numbers \(R_{\frac{k}{2}}(4)=R(4,4,\ldots ,4)\) (resp. \(R_{\frac{k}{2}}(3)\)). We state explicitly these lower bounds for \(k\le 10\) 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 \(t\ge 2\), and for \(R_{t}(3)\), for all \(t \ge 3\).