盒性和分离维数
Boxicity and separation dimension
摘要：
一个超图$H$的顶点置换族$\mathcal{F}$被称为“成对适合”于$H$，如果对于$H$中的每对不相交的边，在$\mathcal{F}$中存在一个置换，其中一条边中的所有顶点都先于另一条边中的顶点. $H$的最小置换族的基数被称为$H$的“分离维数”，用$π(H)$表示。等价地，$π(H)$是最小的自然数$k$，使得$H$的顶点可以嵌入$\mathbb{R}^k$中，使得$H$的任何两个不相交的边可以被垂直于其中一个轴的超平面分开。我们证明了超图$H$的分离维数等于$H$的线图的“盒度”。这种联系有助于我们从大量的关于盒性的文献中借用结果和技术来研究分离维的概念。
英文摘要：
A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $π(H)$. Equivalently, $π(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.
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