TY - JOUR

T1 - A Crossing Lemma for Jordan curves

AU - Pach, János

AU - Rubin, Natan

AU - Tardos, Gábor

N1 - Funding Information:
Supported by the “Lendület” Project in Cryptography of the Hungarian Academy of Sciences and the National Research, Development and Innovation Office — NKFIH projects K-116769 and SNN-117879.
Publisher Copyright:
© 2018

PY - 2018/6/20

Y1 - 2018/6/20

N2 - 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 long-standing 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.

AB - 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 long-standing 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.

KW - Arrangements of curves

KW - Combinatorial geometry

KW - Contact graphs

KW - Crossing Lemma

KW - Extremal problems

KW - Separators

UR - http://www.scopus.com/inward/record.url?scp=85047220040&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.03.015

DO - 10.1016/j.aim.2018.03.015

M3 - Article

AN - SCOPUS:85047220040

SN - 0001-8708

VL - 331

SP - 908

EP - 940

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -