Colorful Helly Theorem for Piercing Boxes with Multiple Points

Abstract: Let $H_c=H_c(d,n)$ denote the smallest positive integer such that if we have a collection of families $\mathcal{F}{1}, \dots, \mathcal{F}{H_{c}}$ of axis-parallel boxes in $\mathbb{R}^{d}$ with the property that every colorful $H_{c}$-tuple from the above families can be pierced by $n$ points then there exits an $i\in { 1, \dots, H_{c}}$, and for all $k\in { 1, \dots, H_{c}} \setminus{i}$ there exists $F_k\in\mathcal{F}k$ such that the following extended family $\mathcal{F}_i\cup\left{F_k\;|\;k\in {1, \dots, H{c}}\;\mbox{and} \;k\neq i\right}$ can also be pierced by $n$ points. In this paper, we give a complete characterization of $H_{c}(d,n)$ for all values of $d$ and $n$. Our result is a colorful generalization of piercing axis-parallel boxes with multiple points by Danzer and Gr\"unbaum (Combinatorica 1982).