An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

Authors Eyal Ackerman, Balázs Keszegh , Günter Rote

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Author Details

Eyal Ackerman
  • Department of Mathematics, Physics, and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel
Balázs Keszegh
  • Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Hungary
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Budapest, Hungary
Günter Rote
  • Department of Computer Science, Freie Universität Berlin, Takustr. 9, 14195 Berlin, Germany


We thank the reviewers for helpful suggestions.

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Eyal Ackerman, Balázs Keszegh, and Günter Rote. An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ⌈ n/6 ⌉), for m ≥ n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatoric problems
  • Simple polygon
  • Ramsey theory
  • combinatorial geometry


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