AI 中文总结
该研究针对整数n阶方阵环，探讨例外团的最大规模，证明部分n的最大例外团非交换，给出无穷多n的非交换例外团规模及所有n的交换例外团规模结果。
AI 中文摘要
我们研究例外团的概念，它是环的一个子集，使得该子集内任意两个不同元素的差都是可逆的。受密码学应用的启发，我们的主要研究重点是确定整数上的n阶方阵组成的环$Mat_{n\times n}(\mathbb{Z})$中，对每个n而言例外团的最大规模。针对上述问题，我们在一般情况和额外要求团中元素两两可交换的“交换”情况中都获得了若干结果。作为亮点，我们证明了至少对部分n值，$Mat_{n\times n}(\mathbb{Z})$中的最大例外团必然是非交换的；随后我们证明，对无穷多的n值，存在规模为$n^2$的非交换例外团，且对每个n，都存在规模为$\frac{2}{3}n+O(n^{\theta})$的交换例外团，其中常数$\theta>\frac{11}{20}$。
英文摘要
We study the notion of exceptional clique, a subset of a ring such that the difference of any two distinct elements of the subset is invertible. Motivated by applications in cryptography, our main focus is to determine the largest size of an exceptional clique in the ring $Mat_{n\times n}(\mathbb{Z})$ of square $n\times n$ matrices over the integers, for every $n$. We obtain several results for the question above, both in the general case and the ``commutative'' case where we additionally require that the elements in the clique commute with each other. As highlights, we prove that, at least for some values of $n$, the largest exceptional cliques in $Mat_{n\times n}(\mathbb{Z})$ are necessarily non-commutative; we then show that for an infinite family of $n$, there are non-commutative exceptional cliques of size $n^2$, and that for every $n$ there are commutative exceptional cliques of size $\frac23 n+O(n^θ)$, for a constant $θ>\frac{11}{20}$.
