Pach, János and Rubin, Natan and Tardos, Gábor (2018) A Crossing Lemma for Jordan curves. ADVANCES IN MATHEMATICS, 331. pp. 908940. ISSN 00018708

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Abstract
If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If T>cn, for some fixed constant c>0, then we prove that X=Ω(T(loglog(T/n))1/504). In particular, if T/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following longstanding conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n2. © 2018
Item Type:  Article 

Uncontrolled Keywords:  Separators; Extremal problems; Crossing lemma; Contact graphs; Combinatorial geometry; Arrangements of curves 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika Q Science / természettudomány > QA Mathematics / matematika > QA166QA166.245 Graphs theory / gráfelmélet Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  16 Aug 2018 13:58 
Last Modified:  31 Mar 2023 11:26 
URI:  http://real.mtak.hu/id/eprint/82752 
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