关于稠密几何图的k色交叉比的一个注记
A Note on the $k$-colored Crossing Ratio of Dense Geometric Graphs
AI总结：
本文证明每个稠密几何图的边可用k种颜色着色，使同色相交边对比例至多为1/k减去某正常数，推广了Aichholzer等人关于完全几何图k=2的结果。
摘要：
几何图是指其顶点集为平面中处于一般位置的点集，且边为连接这些点的直线段的图。我们证明，对于每个整数 k ≥ 2，存在常数 c > 0，使得以下结论成立：每个稠密几何图的边都可以用 k 种颜色着色，使得同色边对中相交的边对数量至多为相交边对总数的 (1/k - c) 倍。当 k = 2 且 G 为完全几何图的情形已由 Aichholzer 等人证明 [GD 2019]。
英文摘要：
A \emph{geometric graph} is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \ge 2$, there exists a constat $c>0$ such that the following holds. The edges of every dense geometric graph can be colored with $k$ colors, such that the number of pairs of edges of the same color that cross is at most $(1/k-c)$ times the total number of pairs of edges that cross. The case when $k=2$ and $G$ is a complete geometric graph, was proved by Aichholzer et al.[\emph{GD} 2019].
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。
