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 1 2 - 7 50 of the total number of crossings.