Lower bounds for piercing and coloring boxes

TL;DR

This paper establishes new lower bounds for the piercing and coloring numbers of axis-parallel boxes in higher dimensions, matching or nearly matching known upper bounds and advancing understanding of geometric intersection graphs.

Contribution

It constructs families of boxes that achieve near-optimal bounds for piercing and coloring numbers, resolving longstanding questions in geometric combinatorics.

Findings

Constructs families of boxes with large piercing numbers relative to independence numbers.

Provides intersection graphs with clique and independence numbers close to theoretical bounds.

Develops bounds on chromatic numbers of box intersection graphs for various clique sizes.

Abstract

Given a family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, let $\tau$ denote its piercing number, and $\nu$ its independence number. It is an old question whether $\tau/\nu$ can be arbitrarily large for given $d\geq 2$. Here, for every $\nu$, we construct a family of axis-parallel boxes achieving $$\tau\geq \Omega_d(\nu)\cdot\left(\frac{\log \nu}{\log\log \nu}\right)^{d-2}.$$ This not only answers the previous question for every $d\geq 3$ positively, but also matches the best known upper bound up to double-logarithmic factors. Our main construction has further implications about the Ramsey and coloring properties of configurations of boxes as well. We show the existence of a family of $n$ boxes in $\mathbb{R}^{d}$, whose intersection graph has clique and independence number $O_d(n^{1/2})\cdot \left(\frac{\log n}{\log\log n}\right)^{-(d-2)/2}.$ This is the first improvement over the trivial upper bound $O_d(n^{1/2})$, and matches the best known lower bound up to double-logarithmic factors. Finally, for every $\omega$ satisfying $\frac{\log n}{\log\log n}\ll \omega\ll n^{1-\varepsilon}$, we construct an intersection graph of $n$ boxes with clique number at most $\omega$, and chromatic number $\Omega_{d,\varepsilon}(\omega)\cdot \left(\frac{\log n}{\log\log n}\right)^{d-2}.$ This matches the best known upper bound up to a factor of $O_d((\log w)(\log \log n)^{d-2})$.