Nerve-type and invariance theorems for asymptotic dimension

2026-07-27

AI Summary

This study investigates the asymptotic dimension of intersection graphs generated by families of sets in metric spaces satisfying diameter and intersection conditions, and establishes quantitative relationships with the Assouad–Nagata dimension of the ambient space. By leveraging geometric covering arguments and dimension theory, the authors provide the first upper bound linking these two dimensions and demonstrate that the asymptotic dimension of such intersection graphs is invariant under boundary effects. Key contributions include showing that the asymptotic dimension of intersection graphs of compact convex sets in ℝⁿ with bounded aspect ratio—such as families of balls—is at most n+1, and that for families of spheres in ℝⁿ (n≥2), the asymptotic dimension is exactly n or n+1. These results are optimal in several respects.

Abstract

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 $n \geq 2$.

Problem

• asymptotic dimension
• intersection graph
• Assouad-Nagata dimension
• metric space
• covering dimension

Innovation

• asymptotic dimension
• Assouad-Nagata dimension
• intersection graph
• metric space
• covering dimension