Stabbing boxes with finitely many axis-parallel lines and flats

TL;DR

This paper characterizes when an infinite collection of axis-parallel boxes in Euclidean space can be intersected by finitely many axis-parallel flats, extending to colorful variants and relating to the Hadwiger-Debrunner problem.

Contribution

It provides the necessary and sufficient conditions for piercing infinite collections of boxes with finitely many flats, including colorful generalizations.

Findings

Established conditions for piercing boxes with flats

Extended results to colorful collections

Linked to the infinite Hadwiger-Debrunner problem

Abstract

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p, q)$-problem.