相邻盒子与二分覆盖；构造与猜想
Neighborly boxes and bipartite coverings; constructions and conjectures
AI总结：
本文研究$\mathbb{R}^d$中两两$k$-相邻盒子的最大数量$n(k,d)$，针对$k=2$提出递归构造，给出下界并猜想其最优性，得到显式公式，由数值实验支持。
摘要：
在$\mathbb{R}^d$中，两个轴对齐的盒子若其交集维度至少为$d-k$且至多为$d-1$，则称它们为$k$-相邻的。在$\mathbb{R}^d$中两两$k$-相邻的盒子的最大数量记为$n(k,d)$。已知对于固定的$1\leqslant k\leqslant d$，$n(k,d)=Θ(d^k)$，但精确公式仅在三种情况下已知：$k=1$，$k=d-1$和$k=d$。特别地，公式$n(1,d)=d+1$等价于Graham和Pollak关于团的二分划分的著名定理。本文处理$k=2$的情况。我们给出了$k$-相邻码的新构造，为$n(2,d)$提供了更好的下界。该构造本质上是递归的，并使用了一种基于三元字符串列表的“代数”，这些列表以熟悉的方式编码相邻盒子。此外，我们猜想我们的构造是最优的，并给出了$n(2,d)$的显式公式。这一猜想得到了一些数值实验和相关开放问题部分结果的支持，这些结果在文中被回顾。
英文摘要：
Two axis-aligned boxes in $\mathbb{R}^d$ are \emph{$k$-neighborly} if their intersection has dimension at least $d-k$ and at most $d-1$. The maximum number of pairwise $k$-neighborly boxes in $\mathbb{R}^d$ is denoted by $n(k,d)$. It is known that $n(k,d)=Θ(d^k)$, for fixed $1\leqslant k\leqslant d$, but exact formulas are known only in three cases: $k=1$, $k=d-1$, and $k=d$. In particular, the formula $n(1,d)=d+1$ is equivalent to the famous theorem of Graham and Pollak on bipartite partitions of cliques. In this paper we are dealing with the case $k=2$. We give a new construction of $k$-neighborly \emph{codes} giving better lower bounds on $n(2,d)$. The construction is recursive in nature and uses a kind of ``algebra'' on \emph{lists} of ternary strings, which encode neighborly boxes in a familiar way. Moreover, we conjecture that our construction is optimal and gives an explicit formula for $n(2,d)$. This supposition is supported by some numerical experiments and some partial results on related open problems which are recalled.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
