Piercing boxes and products of trees

Abstract

We give a new upper bound on the piercing number $τ(\mathcal{F})$ of a finite family $\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $τ(\mathcal{F}) = O(p(\log\log p)(\log\log\log p))$, improving the factor $(\log\log p)^2$ in the best previously known bound to $(\log\log p)(\log\log\log p)$. For each $d>2$, when $\mathcal{F}$ is a finite family of axis-parallel boxes in $\mathbb{R}^d$, this yields $τ(\mathcal{F}) = O(p(\log p)^{d-2}(\log\log p)(\log\log\log p))$. We also study an analogous problem replacing $\mathbb{R}^d=\mathbb{R}\times\cdots\times\mathbb{R}$ with the Cartesian product $T_1\square\cdots\square T_d$ of trees $T_1,\ldots,T_d$, and boxes with objects of the form $S_1\square\cdots\square S_d$ where $S_i$ is a subtree of $T_i$. For a finite family of such objects under the same intersection assumption as above, we establish an upper bound $τ(\mathcal{F})=O(p(\log p)^{d-1})$ for each $d$. We also construct examples showing that this upper bound cannot be improved in order of magnitude.