A note on the 2-colored rectilinear crossing number of random point sets in the unit squareCabello, Sergio (Avtor) Czabarka, Éva (Avtor) Fabila-Monroy, Ruy (Avtor) Higashikawa, Yuya (Avtor) Seidel, Raimund (Avtor) Székely, László (Avtor) Tkadlec, Josef (Avtor) Wesolek, Alexandra (Avtor) arrangement of pointsflathyperplaneLet $S$ 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. 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 ${1 \over 2} - {7 \over 50}$ of the total number of crossings.20242024-10-03 09:46:45Članek v reviji163170sl