Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II

TL;DR

This paper provides a new, simpler proof of the Van Lint--MacWilliams' conjecture related to maximum cliques in Paley graphs, extending previous results to broader Cayley graphs with minimal number-theoretic tools.

Contribution

It introduces a novel, simplified proof of Blokhuis' theorem and its extensions, generalizing to Cayley graphs with small multiplicative doubling.

Findings

New proof of Blokhuis' theorem and extensions

Generalization to Cayley graphs with small doubling

Avoids heavy number-theoretic machinery

Abstract

The well-known Van Lint--MacWilliams' conjecture states that if $q$ is an odd prime power, and $A\subseteq \mathbb{F}_{q^2}$ such that $0,1 \in A$, $|A|=q$, and $a-b$ is a square for each $a,b \in A$, then $A$ must be the subfield $\mathbb{F}_q$. This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if $S \subseteq \mathbb{F}_{q^2}^*$ has small multiplicative doubling, and $A\subseteq \mathbb{F}_{q^2}$ with $0,1 \in A$, $|A|=q$, such that $A-A \subseteq S \cup \{0\}$, then $A=\mathbb{F}_q$. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.