A note on the k-colored crossing ratio of dense geometric graphs

A 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 constant $c>0$ such that the following holds. The edges of every dense geometric graph, with sufficiently many vertices, 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. (2019) [2].