AI 中文总结
本文研究多维0-1矩阵中避开k星时1元素的最大数量，得到了任意d和k的渐近解、k=d的紧边界及d=k=3的精确解，还关联了离散数学多个领域。
AI 中文摘要
一个大小为n₁×n₂×…×n_d的d维0-1矩阵M可被视为布尔函数M: B(n₁×n₂×…×n_d)→{0,1}，其中B是由满足0≤x_i≤n_i-1（1≤i≤d）的格点(x₁,…,x_d)构成的d维盒。该0-1矩阵M也可被描述为B的子集P:=P(M)，满足x∈P当且仅当M(x)=1。M中以p为中心的k星对应于B的一个(k+1)元子集{p,p₁,…,p_k}，其中p与p_i仅在一个坐标上不同（对所有1≤i≤k）且这k个坐标互不相同。本文研究的问题是：确定大小为n×n×…×n的d维0-1矩阵M中，避开所有k星时1元素的最大数量。主要结果包括：对任意d和k，当n→∞时该问题的渐近解；k=d时的紧边界；以及d=k=3情形的精确解。该问题与离散数学的多个其他领域存在关联，包括k部超图、独立集问题、控制集问题和覆盖码。本文的一个工具（关于给定超图诱导副本的最大填充）可能具有独立研究价值。
英文摘要
A $d$-dimensional $0$-$1$ matrix $M$ of size $n_1\times n_2\times \dots \times n_d$ can be considered as a Boolean function $M: B(n_1\times n_2\times \dots \times n_d) \to \{ 0,1\}$, where $B$ is the $d$-dimensional box of lattice points $(x_1, \dots, , x_d)\in Z^d$ with $0\leq x_i \leq n_i-1$, $1\leq i\leq d$. The $0$-$1$ matrix $M$ can also be described as a subset $P:=P(M)$ of $B$ such that $x\in P$ if and only if $M(x)=1$. A $k$-star with center $p$ in $M$ corresponds to a $(k+1)$-element subset $\{ p, p_1, \dots , p_k\} \subset B$ such that $p$ and $p_i$ differ only in one coordinate (for all $1\leq i\leq k$) and these $k$ coordinates are distinct. Here we consider the problem of determining the maximum number of $1$-entries of a $0$-$1$ matrix $M$ of dimension $d$ and size $n\times n \times \dots \times n$ that avoids all $k$-stars. Our main results are the asymptotical solution of the problem for every $d$ and $k$ (as $n\to \infty$), very close bounds for $k=d$, and the exact solution of the $d=k=3$ case. This problem has connections to several other areas of discrete mathematics, including $k$-partite hypergraphs, independent set problems, dominating set problems and covering codes. One of our tools (concerning maximal packings of induced copies of a given hypergraph) might have independent interest.
