Beyond the Richter-Thomassen conjecture
János Pach, Natan Rubin, Gábor Tardos · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 2016
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 fK(loglogn)1/8). As a corollary, we prove 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(l))n2.