关于平面胞腔剖分的等点分离
On Equal Point Separation by Planar Cell Decompositions
摘要：
在本文中，我们调查的问题，分离一组$X$的点在$\mathbb{R}^{2}$与安排的$K$线，使每个细胞包含一个渐近相等的点数（到一个常数比）。我们考虑一个属性的曲线称为刺数，定义为最大可数的交叉点之间的曲线和一条线在平面上。我们发现，大子集的$X$躺在约旦曲线的低刺数是一个障碍，平等的分离。进一步讨论了含有$X$的最小刺数Jordan曲线.我们的结果推广了Erdens-Szekeres猜想的最新界，证明了对于固定的$d$和足够大的$n$，如果$|X| \ge 2^{c_dn/d + o(n)}$与$c_d = 1 + O(\frac{1}{\sqrt{d}})$，则存在$n$点的子集位于Jordan曲线上，刺数至多为$d$ .
英文摘要：
In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a property of curves called the stabbing number, defined to be the maximum countable number of intersections possible between the curve and a line in the plane. We show that large subsets of $X$ lying on Jordan curves of low stabbing number are an obstacle to equal separation. We further discuss Jordan curves of minimal stabbing number containing $X$. Our results generalize recent bounds on the Erdős-Szekeres Conjecture, showing that for fixed $d$ and sufficiently large $n$, if $|X| \ge 2^{c_dn/d + o(n)}$ with $c_d = 1 + O(\frac{1}{\sqrt{d}})$, then there exists a subset of $n$ points lying on a Jordan curve with stabbing number at most $d$.
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