Generalized Paley graphs and their complete subgraphs of orders three and four


Let \(k \ge 2\) be an integer. Let q be a prime power such that \(q \equiv 1 ({\mathrm{mod}}\,\,k)\) if q is even, or, \(q \equiv 1 ({\mathrm{mod}}\,\,2k)\) if q is odd. The generalized Paley graph of order q, \(G_k(q)\), is the graph with vertex set \(\mathbb {F}_q\) 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 \(G_k(q)\), \(\mathcal {K}_4(G_k(q))\), 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 \(G_k(q)\), \(\mathcal {K}_3(G_k(q))\). 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 diagonal Ramsey numbers \(R_k(4)=R(4,4,\ldots ,4)\) (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 \(\mathcal {K}_4(G_k(q))\) and \(\mathcal {K}_3(G_k(q))\), when q is prime, and Fourier coefficients of modular forms.

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Correspondence to Madeline Locus Dawsey.

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The first author was supported by an AMS-Simons travel grant from the American Mathematical Society and the Simons Foundation. The second author was supported by a grant from the Simons Foundation (#353329, Dermot McCarthy).

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Dawsey, M.L., McCarthy, D. Generalized Paley graphs and their complete subgraphs of orders three and four. Res Math Sci 8, 18 (2021).

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Mathematics Subject Classification

  • Primary 05C30
  • 11T24
  • Secondary 05C55
  • 11F11