TY - JOUR

T1 - On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

AU - Pach, János

AU - Rubin, Natan

AU - Tardos, Gábor

N1 - Funding Information: Supported by OTKA grant NN-102029 under EuroGIGA projects GraDR and ComPoSȩ and by Swiss National Science Foundation Grants 200020-144531, 200021-137574 and 200020-162884. Supported by Grant 1452/15 from the Israel Science Foundation and by Grant No 2014384 from the United States-Israel Binational Science Foundation (BSF). Work on this paper by N. R. was partly performed at Université Pierre & Marie Curie and Université Paris Diderot, Institut de Mathématiques de Jussieu (UMR 7586 du CNRS), 4 Place Jussieu, 75252 Paris Cedex, Francȩ supported by the Fondation Sciences Mathématiques de Paris (FSMP) and by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0098). Supported by EPFL, Lausannȩ the "Lendület" programme of the Hungarian Academy of Sciences and the Hungarian OTKA grants NN-102029 and K-116769. Publisher Copyright: © 2016 Cambridge University Press.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the planȩ so that any pair of them intersect and no three curves pass through the same point, is at least (1-o(1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Ω(nt log t/log log t).

AB - A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the planȩ so that any pair of them intersect and no three curves pass through the same point, is at least (1-o(1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Ω(nt log t/log log t).

UR - http://www.scopus.com/inward/record.url?scp=84965071487&partnerID=8YFLogxK

U2 - https://doi.org/10.1017/S0963548316000043

DO - https://doi.org/10.1017/S0963548316000043

M3 - Article

SN - 0963-5483

VL - 25

SP - 941

EP - 958

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 6

ER -