Extremal results on intersection graphs of boxes in $R^d$

TL;DR

This paper determines the maximum number of intersecting pairs among boxes in  with no point common to more than k boxes, improving bounds for the fractional Helly theorem and providing new proofs using advanced combinatorial methods.

Contribution

It provides exact extremal counts for intersection graphs of boxes in  and enhances the understanding of fractional Helly and Erdos-Stone theorems for such geometric configurations.

Findings

Exact maximum number of intersecting pairs in  for boxes with no (k+1)-fold intersections.

Improved bounds for the fractional Helly theorem for boxes.

A second proof of the fractional Helly theorem using advanced combinatorial results.

Abstract

The main purpose of this paper is to study extremal results on the intersection graphs of boxes in $\R^d$. We calculate exactly the maximal number of intersecting pairs in a family $\F$ of $n$ boxes in $\R^d$ with the property that no $k+1$ boxes in $\F$ 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 Erd\H{o}s-Stone theorem for semi-algebraic graphs in order to obtain a second proof of the fractional Helly theorem for boxes.