20.500.12556/DiRROS-20508 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 Ni določen Neznano 2024-10-03 09:42:34 2024-10-03 09:42:34 2024-10-04 03:35:24 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 B9EB5C7AC15B97E236601B03BBA82579 95495778ed5c22280a3706dcf52a085436b6afc70a2d8ecb14990ff1efef2b01 00428a19-815c-11ef-a5bf-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28906 https://link.springer.com/article/10.1007/s10474-024-01436-9 1 0f4c6b38-815b-11ef-a5bf-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28904 Inštitut za matematiko, fiziko in mehaniko 0 0 0