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 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.

van Lint 和 MacWilliams 的一个著名猜想指出，如果A是${\mathbb{F}}_{q^2}$这样$0,1\in \mathrm{一个}$,$|\mathrm{一个}|=q$， 和$\mathrm{一个}-b$是每个正方形$\mathrm{一个},b\in \mathrm{一个}$, 那么A必须是子字段${\mathbb{F}}_q$. 这个猜想通常用佩利图中的最大团来表述。它首先由 Blokhuis 证明，后来由 Sziklai 扩展到广义 Paley 图。在本文中，我们给出了猜想及其变体的新证明，并展示了 Paley 图的 Erdős-Ko-Rado 属性如何扩展到更大的 Cayley 图族，我们称之为 Peisert 型图。这些结果解决了 Mullin 和 Yip 的猜想。