Fractional discrete Helly for pairs in a family of boxes

Abstract

Given a point set $S$ in ${\mathbb{R}}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Soberón proved a fractional version of Halman's theorem for axis-parallel boxes, showing that every finite family $F$ of axis-parallel boxes in ${\mathbb{R}}^d$ with positive density of $S$-intersecting $(d+1)$-tuples contains an $S$-intersecting subfamily of size linear in $|F|$. We prove that qualitatively the same conclusion can be achieved if the density of $S$-intersecting pairs is sufficiently large.

7 pages