Lower bounds for piercing and coloring boxes

Given a family B of axis-parallel boxes in 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 >= 2. Here, for every nu, we construct a family of axis-parallel boxes achieving tau >= Omega(d)(nu)center dot (log nu/log log nu)(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 R-d, whose intersection graph has clique and independence number O-d(n(1/2)) center dot log n/log log n (-(d-2)/2). This is the first improvement log log n 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 log n/log log n << omega << n(1-epsilon), we construct an intersection graph of n boxes with clique number at most omega, and chromatic number Omega(d,epsilon) (omega) center dot (log / log log n) (d-2). This matches the best known upper bound up to a factor of O-d ((log omega)(log logn)(d-2)). (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses/by/4.0/).