Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 239–272 | Cite as

Empty Convex Hexagons in Planar Point Sets



Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.


Erdős-Szekeres problem Ramsey theory Convex polygons and polyhedra Empty hexagon problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bárány, I., Károlyi, Gy.: Problems and results around the Erdős-Szekeres convex polygon theorem. In: Discrete and Computational Geometry, Tokyo, 2000. Lect. Notes Comput. Sci., vol. 2098. Springer, Berlin (2001) CrossRefGoogle Scholar
  2. 2.
    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005) MATHGoogle Scholar
  3. 3.
    Erdős, P.: Some more problems on elementary geometry. Aust. Math. Soc. Gaz. 5, 52–54 (1978) Google Scholar
  4. 4.
    Erdős, P.: Some applications of graph theory and combinatorial methods to number theory and geometry. In: Algebraic Methods in Graph Theory, vols. I, II, Szeged, 1978. Colloq. Math. Soc. János Bolyai, vol. 25, pp. 137–148. North-Holland, Amsterdam (1981) Google Scholar
  5. 5.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935) Google Scholar
  6. 6.
    Erdős, P., Szekeres, G.: On some extremum problems in elementary geometry. Ann. Univ. Sci. Bp. Eötvös Sect. Math. 3–4, 53–62 (1960/1961) Google Scholar
  7. 7.
    Harborth, H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33, 116–118 (1978) MATHMathSciNetGoogle Scholar
  8. 8.
    Horton, J.D.: Sets with no empty convex 7-gons. Can. Math. Bull. 26, 482–484 (1983) MATHMathSciNetGoogle Scholar
  9. 9.
    Morris, W., Soltan, V.: The Erdős-Szekeres problem on points in convex position—a survey. Bull. Am. Math. Soc. 37, 437–458 (2000) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Overmars, M.: Finding sets of points without empty convex 6-gons. Discrete Comput. Geom. 29, 153–158 (2003) MATHMathSciNetGoogle Scholar
  11. 11.
    Tóth, G., Valtr, P.: The Erdős-Szekeres theorem: upper bounds and related results. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 557–568. Cambridge University Press, Cambridge (2005) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany

Personalised recommendations