Asymptotic dimension of intersection graphs

TL;DR

This paper establishes an upper bound on the asymptotic dimension of intersection graphs of certain convex sets in Euclidean space and more general metric spaces, linking geometric properties to combinatorial complexity.

Contribution

It generalizes the bound on asymptotic dimension to intersection graphs of subsets in metric spaces with bounded Assouad-Nagata dimension under specific intersection conditions.

Findings

Asymptotic dimension of intersection graphs is at most 2n+1 for convex sets in R^n.

The result extends to metric spaces with bounded Assouad-Nagata dimension.

Provides a geometric-combinatorial link for intersection graph complexity.

Abstract

We show that intersection graphs of compact convex sets in R^n of bounded aspect ratio have asymptotic dimension at most 2n+1. More generally, we show this is the case for intersection graphs of systems of subsets of any metric space of Assouad-Nagata dimension n that satisfy the following condition: For each r,s>0 and every point p, the number of pairwise-disjoint elements of diameter at least s in the system that are at distance at most r from p is bounded by a function of r/s.