Extensions of discrete Helly theorems for boxes

TL;DR

This paper extends Helly-type theorems for axis-parallel boxes in high-dimensional spaces, introducing colorful, fractional, and quantitative variants, and generalizing to H-convex sets.

Contribution

It provides new colorful, fractional, and quantitative extensions of Halman's discrete Helly theorem, including matroid-based variants and generalizations to H-convex sets.

Findings

Established colorful versions of the theorem with matroid conditions.

Developed fractional versions requiring checks on many (d+1)-tuples.

Generalized results to H-convex sets beyond 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.