Beyond the Richter-Thomassen Conjecture

János Pach, Natan Rubin, Gábor Tardos
2015 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms  
If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point. All other intersection points are called crossing points. The main result of this paper is a Crossing Lemma for closed curves: In any family of n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of at least Ω((log log n) 1/8 ). As a corollary, we
more » ... the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least (1 − o(1))n 2 . * EPFL, Lausanne and Rényi Institute, Budapest.
doi:10.1137/1.9781611974331.ch68 dblp:conf/soda/PachRT16 fatcat:pd3tbur2pzf7vm7s5di6shjkui