A random model for the Paley graph

TL;DR

This paper introduces a new probabilistic model for Paley graphs that captures their complex clique number behavior, including the Graham-Ringrose lower bounds, which previous models failed to reflect.

Contribution

The paper proposes a novel random model incorporating multiplicative structure, accurately reflecting the clique number phenomena of Paley graphs across primes.

Findings

Almost surely, for infinitely many primes, the clique number is ( ext{log} p ext{} ext{log} ext{} ext{p})

For most primes, the clique number is approximately 2 log p

The model captures the Graham-Ringrose lower bounds for clique numbers

Abstract

For a prime $p$ we define the Paley graph to be the graph with the set of vertices $\mathbb{Z}/p\mathbb{Z}$, and with edges connecting vertices whose sum is a quadratic residue. Paley graphs are notoriously difficult to study, particularly finding bounds for their clique numbers. For this reason, it is desirable to have a random model. A well known result of Graham and Ringrose shows that the clique number of the Paley graph is $\Omega(\log p\log\log\log p)$ (even $\Omega(\log p\log\log p)$, under the generalized Riemann hypothesis) for infinitely many primes $p$ -- a behaviour not detected by the random Cayley graph which is hence deficient as a random model for for the Paley graph. In this paper we give a new probabilistic model which incorporates some multiplicative structure and as a result captures the Graham-Ringrose phenomenon. We prove that if we sample such a random graph independently for every prime, then almost surely (i) for infinitely many primes $p$ the clique number is $\Omega(\log p\log\log p)$, whilst (ii) for almost all primes the clique number is $(2+o(1))\log p$.