Extremal Results on Intersection Graphs of Boxes in ${\mathbb R}^d$

Abstract

The main purpose of this paper is to study extremal results on the intersection graphs of boxes in ${\mathbb R}^d$. We calculate exactly the maximal number of intersecting pairs in a family ${\mathcal F}$ of n boxes in ${\mathbb R}^d$ with the property that no $k+1$ boxes in ${\mathcal F}$ have a point in common. This allows us to improve the known bounds for the fractional Helly theorem for boxes. We also use the Fox–Gromov–Lafforgue–Naor–Pach results to derive a fractional Erdős–Stone theorem for semi-algebraic graphs in order to obtain a second proof of the fractional Helly theorem for boxes.

Dedicated to Tudor Zamfirescu.