广义Paley图及其三阶和四阶完全子图
Generalized Paley graphs and their complete subgraphs of orders three and four
摘要：
令$k \geq 2$为整数。令$q$为素数幂，使得如果$q$为偶数，则为$q \equiv 1 \pmod {k}$，或者如果$q$为奇数，则为$q \equiv 1 \pmod {2k}$。$q$阶广义Paley图$G_k(q)$是顶点集为$\mathbb{F}_q$的图，其中$ab$是边当且仅当${a-b}$是$k$次幂剩余.我们提供了一个公式，在有限域超几何函数，$G_k(q)$，$\mathcal{K}_4(G_k(q))$中包含的四阶完全子图的数量，它适用于所有的$k$。这推广了Evans，Pulham和Sheehan关于原始（$k$ =2）Paley图的结果.我们还提供了一个公式，在雅可比和方面，$G_k(q)$，$\mathcal{K}_3(G_k(q))$中包含的三阶完全子图的数量。在这两种情况下，我们给出明确的决定，这些公式为小$k$。我们表明，零值的$\mathcal{K}_4(G_k(q))$（分别）。$\mathcal{K}_3(G_k(q))$）给出了双对角Ramsey数$R_k(4)=R(4,4,\cdots,4)$（分别为$R_k(3)$）。我们明确指出了小$k$的这些下限，并与已知的界限进行比较。我们还检查之间的关系$\mathcal{K}_4(G_k(q))$和$\mathcal{K}_3(G_k(q))$，当$q$是素数，和模形式的傅立叶系数。
英文摘要：
Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1 \pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ 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 $k$-th 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 $R_k(4)=R(4,4,\cdots,4)$ (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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