标题： 范·林特-马卡威廉姆斯猜想与有限域上Cayley图的最大团，II 显示英文标题

标题： Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II

摘要： 著名的范·林特-麦克威廉姆斯猜想指出，如果 $q$ 是奇素数幂，且 $A\subseteq \mathbb{F}_{q^2}$ 满足 $0,1 \in A$, $|A|=q$, 和 $a-b$ 对每个 $a,b \in A$ 都是平方数，则 $A$ 必须是子域 $\mathbb{F}_q$。 这一猜想最初由Blokhuis证明，通常表述为关于平方阶Paley图的最大团的问题。此前，Asgarli和作者将Blokhuis的定理扩展到了更大的Cayley图族。本文给出了Blokhuis定理及其扩展的一个新的简单证明。更一般地，我们证明了如果 $S \subseteq \mathbb{F}_{q^2}^*$ 的乘法加倍较小，并且存在 $A\subseteq \mathbb{F}_{q^2}$, $0,1 \in A$, $|A|=q$，使得 $A-A \subseteq S \cup \{0\}$，则有 $A=\mathbb{F}_q$。这一新结果改进并推广了多个先前的工作；此外，我们的新方法避免使用数论中的繁重工具。 显示英文摘要

摘要： The well-known Van Lint--MacWilliams' conjecture states that if $q$ is an odd prime power, and $A\subseteq \mathbb{F}_{q^2}$ 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 was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if $S \subseteq \mathbb{F}_{q^2}^*$ has small multiplicative doubling, and $A\subseteq \mathbb{F}_{q^2}$ with $0,1 \in A$, $|A|=q$, such that $A-A \subseteq S \cup \{0\}$, then $A=\mathbb{F}_q$. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.