Extensions of discrete Helly theorems for boxes

Abstract

We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of any $2d$ contains a point of $S$, then the intersection of $F$ contains a point of $S$. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many $(d+1)$-tuples of the family contain points of $S$. Among the colorful versions we include variants where the coloring condition is replaced by an arbitrary matroid. Our results generalize beyond axis-parallel boxes to $H$-convex sets.