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 - 10.1137/1.9781611974331.ch68
DO - 10.1137/1.9781611974331.ch68
M3 - Conference contribution
AN - SCOPUS:84963669727
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
PB - Association for Computing Machinery
T2 - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
Y2 - 10 January 2016 through 12 January 2016
ER -