TY - GEN

T1 - Beyond the Richter-Thomassen conjecture

AU - Pach, János

AU - Rubin, Natan

AU - Tardos, Gábor

N1 - Funding Information: Supported by Swiss National Science Foundation Grants 200020-144531 and 20021-137574. Supported by grant 1452/15 from Israel Science Foundation, by grant 2014384 from the U.S.-Israeli Binational Science Foundation, by the Frenkel Foundation, by the Fondation Sciences Mathematiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements dAvenir program (reference: ANR-lO-LABX-0098).%blankline%

PY - 2016/1/1

Y1 - 2016/1/1

N2 - If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point-All other intersection points are called 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 fK(loglogn)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(l))n2.

AB - If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point-All other intersection points are called 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 fK(loglogn)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(l))n2.

UR - http://www.scopus.com/inward/record.url?scp=84963669727&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/1.9781611974331.ch68

DO - https://doi.org/10.1137/1.9781611974331.ch68

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 957

EP - 968

BT - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

A2 - Krauthgamer, Robert

T2 - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

Y2 - 10 January 2016 through 12 January 2016

ER -