2-piercings via graph theory (Q1003742)

A \(d\)-dimensional box is a rectangular parallelepiped whose edges are parallel to the coordinate axes. A family of boxes is called \(n\)-pierceable if there is a set of \(n\) points such that each box contains at least one of these points. The paper deals with a particular case of the Helly-type problem: given the positive integers \(d\) and \(n\) what is the smallest number \(h=h(d,n)\) such that a family of boxes in \(d\)-dimensional space is \(n\)-pierceable. The authors give a simple proof of the following theorem of \textit{L. Danzer} and \textit{B. Grünbaum} [Combinatorica 2, 237--246 (1982; Zbl 0513.52009)]: \(h(d,2)=3d\) for odd \(d\), and \(h(d,2)=3d-1\) for even \(d\). They turn the problem to analysing the structure of odd anti-holes as an element in the lower bound original construction, showing that it does not play a role in the main argument.