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≥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.