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

TL;DR

This paper derives formulas for counting complete subgraphs of size three and four in generalized Paley graphs using finite field hypergeometric functions and Jacobi sums, extending previous results and exploring their implications for Ramsey numbers.

Contribution

It provides new explicit formulas for subgraph counts in generalized Paley graphs for all k, generalizing prior work on the classical Paley graph, and connects these counts to Ramsey bounds and modular forms.

Findings

Formulas for $\

Formulas for $\

Explicit bounds for multicolor diagonal Ramsey numbers based on subgraph counts

Abstract

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.