低维欧几里得空间中半空间间的匹配与碰集
Matchings vs hitting sets among half-spaces in low dimensional euclidean spaces
摘要：
令$\mathcal{F}$是在$\mathbb{R}^2$或$\mathbb{R}^3$中的$n$点的集合$P$的线性可分集合的任何集合。我们发现，对于每一个自然数$k$可以找到$k$成对不相交的集在$\mathcal{F}$，或有$O(k)$点在$P$一起击中所有集在$\mathcal{F}$。证明是基于显示类似的结果家庭$\mathcal{F}$可分的伪盘在$\mathbb{R}^2$。我们补充这些陈述表明，类似的结果不成立的集合的线性可分集$\mathbb{R}^4$和高维欧几里得空间。
英文摘要：
Let $\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\mathbb{R}^2$, or in $\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in $\mathcal{F}$, or there are $O(k)$ points in $P$ that together hit all sets in $\mathcal{F}$. The proof is based on showing a similar result for families $\mathcal{F}$ of sets separable by pseudo-discs in $\mathbb{R}^2$. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in $\mathbb{R}^4$ and higher dimensional euclidean spaces.
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