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

TL;DR

This paper provides a new proof of the van Lint-MacWilliams conjecture related to maximum cliques in Paley graphs and extends the Erdős-Ko-Rado property to Peisert-type Cayley graphs, resolving existing conjectures.

Contribution

It introduces a novel proof of the van Lint-MacWilliams conjecture and extends the Erdős-Ko-Rado property to a broader class of Cayley graphs, confirming conjectures by Mullin and Yip.

Findings

New proof of the van Lint-MacWilliams conjecture

Extension of Erdős-Ko-Rado property to Peisert-type graphs

Resolution of Mullin and Yip's conjectures

Abstract

A well-known conjecture due to van Lint and MacWilliams states that if $A$ is a subset of $\mathbb{F}_{q^2}$ 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 is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by Blokhuis and later extended by Sziklai to generalized Paley graphs. In this paper, we give a new proof of the conjecture and its variants, and show this Erd\H{o}s-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs, resolving conjectures by Mullin and Yip.