In 1978, van Lint and MacWilliams studied vectors of minimum weight in certain generalized quadratic residue-codes, and they proposed the following conjecture: 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 any $a,b \in A$, then $A$ must be given by the subfield $\mathbb{F}_q$. The conjecture is often phased in terms of the maximum cliques in Paley graphs. The conjecture was first proved by Blokhuis in 1984; in 1999, Sziklai generalized Blokhuis's proof and extended the the conjecture to certain generalized Paley graphs. In this talk, I will present a new proof of this conjecture and its variants, and show that such characterization of maximum cliques extends to a larger family of Cayley graphs, including Peisert graphs, resolving conjectures by Mullin and Yip. This is a joint work with Chi Hoi Yip, who is a PhD student at the University of British Columbia.