Colorful two-piercing theorem for boxes

TL;DR

This paper extends a Helly-type theorem to a colorful setting for two-piercing families of axis-parallel boxes in any dimension, providing tight bounds and related graph-theoretic insights.

Contribution

It introduces a new colorful two-piercing theorem for boxes, extending previous results and proving the bounds are tight through extremal constructions.

Findings

Established a colorful extension of the two-piercing theorem for boxes.

Proved the bounds are tight with extremal examples.

Connected the result to related graph-theoretic proofs.

Abstract

We prove a colorful extension of a Helly-type theorem by Danzer and Gr\"{u}nbaum (Combinatorica, 1982) concerning two-piercing families of axis-parallel boxes in $\mathbb{R}^d$. We also show that our result is tight by constructing extremal families that achieve the bound. Related work includes a graph-theoretic proof of the original theorem by Pendavingh, Puite, and Woeginger (Discrete Applied Mathematics, 2008), and a two-piercing result for lower-dimensional boxes by Ba\~{n}os and Oliveros (Acta Mathematica Hungarica, 2018).