2015/04/30 by János Pach, Pach, János, Natan Rubin +3
Mathematics · #05C10 #05C35 #05D99 #52C10 #52C30 #52C45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #Computational Geometry (cs.CG) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.1 #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1504.08250
openalex publication_date 2015/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If two closed Jordan curves in the plane have precisely one point in common, then it is called a \em touching point. All other intersection points are called \em 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((loglog n)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(1))n2.