交叉族的一个推广
A generalization of crossing families
摘要：
对于平面中的一组点，\emph{crossing family}是一组线段，每个线段连接两个点，使得任何两个线段相交。我们研究了交叉族的以下推广：一个\emph{spoke set}是一组通过点集绘制的线，使得诱导线排列的每个无界区域包含点集的至少一个点。我们证明了每个点集都有一个大小为$\sqrt{\frac{n}{8}}$的辐集。我们还描述了匹配，通过选择一个点，在每个无界区域和连接每个这样的点，在对映无界区域的点。
英文摘要：
For a set of points in the plane, a \emph{crossing family} is a set of line segments, each joining two of the points, such that any two line segments cross. We investigate the following generalization of crossing families: a \emph{spoke set} is a set of lines drawn through a point set such that each unbounded region of the induced line arrangement contains at least one point of the point set. We show that every point set has a spoke set of size $\sqrt{\frac{n}{8}}$. We also characterize the matchings obtained by selecting exactly one point in each unbounded region and connecting every such point to the point in the antipodal unbounded region.
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