关于杂交家系的有关问题
On problems related to crossing families
摘要：
给定平面中的一组点，\emph{crossing family}是线段的集合，每个线段连接两个点，使得每两个线段在内部相交。Aronov等人[Combinatorica，~14（2）：127-134，~1994]证明了任何$n$点集都包含一个大小为$Ω(\sqrt{n})$的交叉族。他们还提到，存在最大交叉族使用最多$\frac{n}{2}$点的点集。我们改进了交叉族规模的上界为$5\lceil \frac{n}{24} \rceil$。我们还介绍了一些推广的交叉家庭，并给出了几个下限和上限我们的广义概念。
英文摘要：
Given a set of points in the plane, a \emph{crossing family} is a collection of segments, each joining two of the points, such that every two segments intersect internally. Aronov et al. [Combinatorica,~14(2):127-134,~1994] proved that any set of $n$ points contains a crossing family of size $Ω(\sqrt{n})$. They also mentioned that there exist point sets whose maximum crossing family uses at most $\frac{n}{2}$ of the points. We improve the upper bound on the size of crossing families to $5\lceil \frac{n}{24} \rceil$. We also introduce a few generalizations of crossing families, and give several lower and upper bounds on our generalized notions.
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。
