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].

翻译：**摘要：** 一个*几何图*是指其顶点集为平面上处于一般位置的点集，且其边为连接这些点的直线段的图。我们证明：对于任意整数$k \ge 2$，存在常数$c>0$，使得以下结论成立。每个稠密几何图的边可用$k$种颜色进行着色，使得同色边交叉的对数不超过总交叉边对数的$(1/k-c)$倍。当$k=2$且$G$为完全几何图时，该结论由Aichholzer等人[《图形与绘图》2019]证明。