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

Abstract

A well-known conjecture due to van Lint and MacWilliams states that if A is a subset of Fq2 such that 0,1∈A, |A|=q, and a−b is a square for each a,b∈A, then A must be the subfield Fq. 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 how this Erdős-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs. These results resolve the conjectures by Mullin and Yip.