Lower bounds for piercing and coloring boxes

Abstract

Given a family B of axis-parallel boxes in Rd, let τ denote its piercing number, and ν its independence number. It is an old question whether τ/ν can be arbitrarily large for given d≥2. Here, for every ν, we construct a family of axis-parallel boxes achievingτ≥Ωd(ν)⋅(log⁡νlog⁡log⁡ν)d−2. This not only answers the previous question for every d≥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 Rd, whose intersection graph has clique and independence number Od(n1/2)⋅(log⁡nlog⁡log⁡n)−(d−2)/2. This is the first improvement over the trivial upper bound Od(n1/2), and matches the best known lower bound up to double-logarithmic factors. Finally, for every ω satisfying log⁡nlog⁡log⁡n≪ω≪n1−ε, we construct an intersection graph of n boxes with clique number at most ω, and chromatic number Ωd,ε(ω)⋅(log⁡nlog⁡log⁡n)d−2. This matches the best known upper bound up to a factor of Od((log⁡ω)(log⁡log⁡n)d−2).