Colorful two-piercing theorem for boxes
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 . 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).
Cite
@article{arxiv.2207.14368,
title = {Colorful two-piercing theorem for boxes},
author = {Sourav Chakraborty and Arijit Ghosh and Soumi Nandi},
journal= {arXiv preprint arXiv:2207.14368},
year = {2025}
}
Comments
Major revision. Large parts of the paper have been rewritten to improve clarity and presentation. The two-piercing result is retained, while the material on three-piercing has been removed. This paper will appear in Discrete Applied Mathematics