A note on the 2-colored rectilinear crossing number of random point sets in the unit square

Let \(S\) be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of \(S\) with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that \(S\) defines a pair of crossing edges of the same color is equal to \(1/4\). This is connected to a recent result of Aichholzer et al. [1] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation \(\frac{1}{2}-\frac{7}{50}\) of the total number of crossings.