arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1006.3191math.CO
收藏

半平面的多色着色

Polychromatic Coloring for Half-Planes

Shakhar Smorodinsky, Yelena Yuditsky

备注：
11 pages, 5 figures
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们证明了对于每一个整数$k$，平面上的每一个有限点集都可以是$k$着色的，使得每一个至少包含$2k-1$点的半平面也至少包含一个来自每一个颜色类的点。我们还表明，绑定的$2k-1$是最好的可能。这分别改进了$\frac{4}{3}k$和$4k-1$的最佳先前已知的下限和上限。我们还表明，每一个有限集的半平面可以$k$着色，使如果一个点$p$属于一个子集$H_p$至少$3k-2$的半平面，然后$H_p$包含一个半平面从每个颜色类。这改进了$8k-3$的最佳先前已知的上界。我们的第一个结果的另一个推论是一个新的证明存在的小尺寸$\eps$ -网的平面上的点相对于半平面。

英文摘要：

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the bound $2k-1$ is best possible. This improves the best previously known lower and upper bounds of $\frac{4}{3}k$ and $4k-1$ respectively. We also show that every finite set of half-planes can be $k$ colored so that if a point $p$ belongs to a subset $H_p$ of at least $3k-2$ of the half-planes then $H_p$ contains a half-plane from every color class. This improves the best previously known upper bound of $8k-3$. Another corollary of our first result is a new proof of the existence of small size $\eps$-nets for points in the plane with respect to half-planes.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑