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.

令\（k \ ge 2 \）为整数。如果q是偶数，则令q为\（q \ equiv 1（{\ mathrm {mod}} \，\，k）\）的素数，或者\（q \ equiv 1（{\ mathrm {mod} } \，\，2k）\）如果q为奇数。顺序的广义佩利图形q，\（G_k（Q）\） ，是与顶点集合的曲线\（\ mathbb {F} _q \） ，其中AB是边缘当且仅当\（{AB} \）是一个ķ次方残基。我们提供了一个基于有限域超几何函数的公式，用于\（G_k（q）\）中包含的四阶完整子图的数量，\（\ mathcal {K} _4（G_k（q））\），它适用于所有k。这将在原始（\（k = 2 \））Paley图上推广Evans，Pulham和Sheehan的结果。我们还根据Jacobi和提供了一个公式，用于\（G_k（q）\），\（\ mathcal {K} _3（G_k（q））\）中包含的三阶完整子图的数量。在这两种情况下，我们都会为小k明确确定这些公式。我们证明\（\ mathcal {K} _4（G_k（q））\）（resp。  \（\ mathcal {K} _3（G_k（q））\））的零值产生多色对角线Ramsey的下限数字\（R_k（4）= R（4,4，\ ldots，4）\）（分别为 \（R_k（3）\））。我们为小k明确声明这些下限，并与已知范围进行比较。当q为素数时，我们还检查了\（\ mathcal {K} _4（G_k（q））\）和\（\ mathcal {K} _3（G_k（q））\）之间的关系，以及模块化形式。