Many Touchings Force Many Crossings

  • János Pach
  • Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)


Given n continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let t and c denote the number of touching pairs and crossing pairs, respectively. We prove that \(c \ge {1\over 10^5}{t^2\over n^2}\), provided that \(t\ge 10n\). Apart from the values of the constants, this result is best possible.


Planar curves Touching Crossing 



The work of János Pach was partially supported by Swiss National Science Foundation Grants 200021-165977 and 200020-162884. Géza Tóth’s work was partially supported by the Hungarian National Research, Development and Innovation Office, NKFIH, Grant K-111827.


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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Rényi InstituteHungarian Academy of SciencesBudapestHungary

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