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29 March 2024
 
  » arxiv » 1504.8250

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Beyond the Richter-Thomassen Conjecture
János Pach ; Natan Rubin ; Gábor Tardos ;
Date 30 Apr 2015
AbstractIf two closed Jordan curves in the plane have precisely one point in common, then it is called a {em touching point}. All other intersection points are called {em 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 at least $Omega((loglog n)^{1/8})$.
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 $(1-o(1))n^2$.
Source arXiv, 1504.8250
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