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arXiv · 1504.08250

Beyond the Richter-Thomassen Conjecture

János Pach·Natan Rubin·Gábor Tardos
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Abstract

If 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 $Ω((\log\log n)^{1/8})$. 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 $(1-o(1))n^2$.

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János Pach, Natan Rubin, Gábor Tardos. 2015-07-07. Beyond the Richter-Thomassen Conjecture. https://arxiv.org/abs/1504.08250

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