论文标题
多点的五颜六色的helly定理，用于多个点的刺穿盒子
Colorful Helly Theorem for Piercing Boxes with Multiple Points
论文作者
论文摘要
令$ h_c = h_c（d，n）$表示最小的积极整数，以便我们有$ \ Mathcal {f} _ {1} _ {1}，\ dots，\ Mathcal {f} _ { $ h_ {c} $ - 可以用$ n $点刺穿的元素，然后在\ {1，\ dots，h_ {c} \} $中退出一个$ i \，以及所有$ k \ in \ in \ in \ {1，\ dots，\ dots，h_ {c} {c} {c} \} \ setMinus \} $ f_k \ in \ nathcal {f} _k $使以下大家庭$ \ MATHCAL {f} _i \ cup \ left \ left \ {f_k \; | \; | \; k \; k \ in \ in \ in \ {1，\ dots，\ dots，h_ _ {c} {c} {c} \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \;也可以用$ n $点刺穿。在本文中，对于所有$ d $和$ n $的值，我们给出了$ h_ {c}（d，n）$的完整表征。我们的结果是Danzer和Grünbaum的多个点的穿孔轴平行盒进行了多彩概括（Combinatorica 1982）。
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 Grünbaum (Combinatorica 1982).
