A well-known conjecture due to van Lint and MacWilliams states that if A is a subset of F q 2 such that 0, 1∈ A,| A|= q, and a− b is a square for each a, b∈ A, then A must be the subfield 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 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.