Paper Info
Title
A Crossing Lemma for Jordan Curves.
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|(log⁡log⁡(|T|/n))1/504). In particular, if |T|/n→∞, then the number of intersection points is much larger than the number of touching points.
Year
DOI
Venue
2017
10.1016/j.aim.2018.03.015
Advances in Mathematics
Keywords
Field
DocType
Extremal problems,Combinatorial geometry,Arrangements of curves,Crossing Lemma,Separators,Contact graphs
Discrete mathematics,Combinatorics,Omega,Corollary,Conjecture,Mathematics,Lemma (mathematics)
Journal
Volume
ISSN
Citations 
331
0001-8708
0
PageRank 
References 
Authors
0.34
19
3
Name
Order
Citations
PageRank
János Pach12366292.28
Natan Rubin29211.03
Gábor Tardos31261140.58