  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.
