A fractional Helly theorem for boxes

TL;DR

This paper establishes a fractional Helly-type theorem for axis-parallel boxes in high-dimensional space, linking the density of pairwise intersections to the existence of large intersecting subfamilies.

Contribution

It proves a new fractional Helly theorem for boxes, showing that a high proportion of intersecting pairs guarantees a large intersecting subfamily, extending Helly-type results.

Findings

If a family of boxes has a high density of intersecting pairs, then a large intersecting subfamily exists.

The threshold for the density of intersecting pairs is sharp, as shown by a constructed example.

The result generalizes classical Helly theorems to a fractional setting for boxes.

Abstract

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