Tur\'an-type results for intersection graphs of boxes

TL;DR

This paper establishes a Turán-type upper bound for the number of edges in intersection graphs of boxes in higher dimensions that exclude complete bipartite subgraphs, linking geometric and combinatorial properties.

Contribution

It proves a new upper bound for intersection graphs of boxes avoiding $K_{t,t}$, connecting boxicity, separation dimension, and poset dimension, and disproves a conjecture on separation dimension.

Findings

Bound on edges: $ctn(\log n)^{2d+3}$ for $K_{t,t}$-free intersection graphs.

Construction of graphs with separation dimension 4 having superlinear edges.

Disproof of a conjecture by Alon et al. using geometric graph constructions.

Abstract

In this short note, we prove the following analog of the K\H{o}v\'ari-S\'os-Tur\'an theorem for intersection graphs of boxes. If $G$ is the intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that $G$ contains no copy of $K_{t,t}$, then $G$ has at most $ctn(\log n)^{2d+3}$ edges, where $c=c(d)>0$ only depends on $d$. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit et al. of $K_{2,2}$-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon et al. We show that there exist graphs of separation dimension 4 having superlinear number of edges.