AI 中文总结
研究度量空间渐近维数与集合族交图的关系，在一定假设下证明了\(\mathcal{F}\)交图渐近维数与环境空间阿苏阿德 - 纳加塔维数的联系，还得出连通拓扑空间中相关集族交图渐近维数的结论，给出了如\(\mathbb{R}^n\)中球族交图渐近维数的结果。
AI 中文摘要
度量空间的渐近维数是拓扑空间覆盖维数的大规模类似物。集合族的交图是一个图，其顶点是集合族的成员，边对应于交集非空的成员对。我们的第一个主要结果在一些温和且必要的假设下，将集合族\(\mathcal{F}\)的交图的渐近维数与包含\(\mathcal{F}\)成员的环境度量空间的阿苏阿德 - 纳加塔维数联系起来。我们证明，如果\(\mathcal{F}\)是阿苏阿德 - 纳加塔维数为\(n\)的度量空间的子集族，使得对于某个函数\(f\)，半径为\(r\)的每个球最多与\(f(r/s)\)个直径至少为\(s\)的两两不相交的\(\mathcal{F}\)成员相交，那么\(\mathcal{F}\)的交图的渐近维数至多为\(n + 1\)。这个结果在几个方面在定量和定性上都是最优的。作为这个结果的推论，\(\mathbb{R}^n\)中任何有界纵横比紧凑凸集族（例如\(\mathbb{R}^n\)中的球族）的交图的渐近维数至多为\(n + 1\)。我们的第二个主要结果表明，在一个温和条件下，具有连通边界的连通拓扑空间中连通闭集族\(\mathcal{F}\)的交图的渐近维数等于\(\mathcal{F}\)中集合边界族的交图的渐近维数。特别地，当\(n\geq2\)时，\(\mathbb{R}^n\)中球族的交图的渐近维数等于\(n\)或\(n + 1\)。
英文摘要
Asymptotic dimension of metric spaces is a large-scale analog of covering dimension of topological spaces. An intersection graph of a family of sets is the graph whose vertices are the members of the family and whose edges correspond to pairs of members with non-empty intersection. Our first main result connects the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ and the Assouad-Nagata dimension of the ambient metric space containing members of ${\mathcal F}$ under some mild and necessary assumptions. We prove that if ${\mathcal F}$ is a family of subsets of a metric space of Assouad-Nagata dimension $n$ such that every ball of radius $r$ intersects at most $f(r/s)$ pairwise disjoint members of ${\mathcal F}$ of diameter at least $s$ for some function $f$, then the asymptotic dimension of the intersection graph of ${\mathcal F}$ is at most $n+1$. This result is optimal both quantitatively and qualitatively in several senses. As a corollary of this result, the asymptotic dimension of the intersection graph of any family of compact convex sets of bounded aspect ratio in ${\mathbb R}^n$, such as a family of balls in ${\mathbb R}^n$, is at most $n+1$. Our second main result states that the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ of connected closed sets of a connected topological space with connected boundary equals the asymptotic dimension of the intersection graph of the family of the boundary of the sets in ${\mathcal F}$, under a mild condition. In particular, the asymptotic dimension of the intersection graphs of families of spheres in ${\mathbb R}^n$ equals $n$ or $n+1$ when $n \geq 2$.
