20.500.12556/RUL-163170 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. 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. arrangement of points flat hyperplane true false true Angleški jezik Angleški jezik Članek v reviji 2024-10-03 09:46:45 2024-10-03 09:46:47 2024-10-08 03:32:15 0000-00-00 00:00:00 2024 0 0 Str. 214-226 iss. 1 Vol. 173 Jun. 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-06-01 514.17 0236-5294 10.1007/s10474-024-01436-9 206428931 s10474-024-01436-9-4.pdf s10474-024-01436-9-4.pdf 1 6D883C44F81E31CED4F756756E08099B 8da881ec3e2ffdd2f91ecf0f0ac474aadaaab8ccf1a3be6ce83a9aa41115aefa e44027ae-815b-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=192687 https://link.springer.com/article/10.1007/s10474-024-01436-9 1 7cfa4160-815b-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=192686 Fakulteta za matematiko in fiziko 0 0 0