Periodica Mathematica Hungarica

, Volume 55, Issue 2, pp 121–127 | Cite as

Empty convex polygons in almost convex sets

  • Pavel Valtr
  • Gábor Lippner
  • Gyula Károlyi


A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every ℓ ≥ 3, we determine the maximum size of an almost convex set that does not contain the vertex set of an empty convex ℓ-gon.

Key words and phrases

point configurations combinatorial convexity Erdős-Szekeres problem empty polygons 

Mathematics subject classification number



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  1. [1]
    P. Erdős, On some problems of elementary and combinatorial geometry, Ann. Mat. Pura Appl. (4), 103 (1975), 99–108.CrossRefMathSciNetGoogle Scholar
  2. [2]
    P. Erdős and Gy. Szekeres, A combinatorial problem in geometry, Compositio Math., 2 (1935), 464–470.Google Scholar
  3. [3]
    T. Gerken, Empty convex hexagons in planar point sets, Discrete Comput. Geom., to appear.Google Scholar
  4. [4]
    H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elem. Math., 33 (1978), 116–118.MATHMathSciNetGoogle Scholar
  5. [5]
    J. D. Horton, Sets with no empty convex 7-gons, Canad. Math. Bull., 26 (1983), 482–484.MATHMathSciNetGoogle Scholar
  6. [6]
    Gy. Károlyi, J. Pach and G. Tóth, A modular version of the Erdős-Szekeres theorem, Studia Sci. Math. Hungar., 38 (2001), 245–259.MATHMathSciNetGoogle Scholar
  7. [7]
    G. Kun and G. Lippner, Large convex empty polygons in k-convex sets, Period. Math. Hungar., 46 (2003), 81–88.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    C. M. Nicolás, The empty hexagon theorem, Discrete Comput. Geom., 38 (2007), 389–397.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P. Valtr, A sufficient condition for the existence of large empty convex polygons, Discrete Comput. Geom., 28 (2002), 671–682.MATHMathSciNetGoogle Scholar
  10. [10]
    P. Valtr, Open caps and cups in planar point sets, Discrete Comput. Geom., 37 (2007), 565–576.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Valtr, On the empty hexagons, submitted.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  2. 2.Institute of MathematicsEötvös UniversityBudapestHungary

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