A fractional Helly theorem for boxes

Abstract

Let $\mathcal{F}$ be a family of $n$ axis-parallel boxes in ${\mathbb{R}}^d$ and $\hat{I}\pm \in (1-1/d,1]$ a real number. There exists a real number ${\hat{I}}^2(\hat{I}\pm )>0$ such that if there are $\hat{I}\pm \left(\genfrac{}{}{0ex}{}{n}{2}\right)$ intersecting pairs in $\mathcal{F}$, then $\mathcal{F}$ contains an intersecting subfamily of size ${\hat{I}}^{2}n$. A simple example shows that the above statement is best possible in the sense that if $\hat{I}\pm \le 1-1/d$, then there may be no point in ${\mathbb{R}}^d$ that belongs to more than $d$ elements of $\mathcal{F}$.