Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Abstract

We prove the following generalised empty pentagon theorem for every integer  ≥ 2, every sufficiently large set of points in the plane contains collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].

This is a preview of subscription content, log in to check access.

References

  1. 1

    Addario-Berry, L., Fernandes, C., Kohayakawa, Y., Coelho de Pina, J., Wakabayashi, Y.: On a geometric Ramsey-style problem (2007). http://crm.umontreal.ca/cal/en/mois200708.html

  2. 2

    Bárány, I., Károlyi, G.: Problems and results around the Erdős-Szekeres convex polygon theorem. In: Proceedings of Japanese Conf. on discrete and computational geometry (JCDCG 2000). Lecture Notes in Comput. Sci., vol. 2098, pp. 91–105. Springer, Berlin (2001)

  3. 3

    Bárány I., Valtr P.: Planar point sets with a small number of empty convex polygons. Studia Sci. Math. Hungar. 41(2), 243–266 (2004)

    MATH  MathSciNet  Google Scholar 

  4. 4

    Bisztriczky T., Hosono K., Károlyi G., Urabe M.: Constructions from empty polygons. Period. Math. Hungar. 49(2), 1–8 (2004). doi:10.1007/s10998-004-0518-7

    MATH  MathSciNet  Article  Google Scholar 

  5. 5

    Braß, P.: On point sets without k collinear points. In: Discrete Geometry. Monogr. Textbooks Pure Appl. Math., vol. 253, pp. 185–192. Dekker, NY (2003)

  6. 6

    Braß P., Moser W.O.J., Pach J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)

    Google Scholar 

  7. 7

    Dobkin D.P., Edelsbrunner H., Overmars M.H.: Searching for empty convex polygons. Algorithmica 5(4), 561–571 (1990). doi:10.1007/BF01840404

    MATH  MathSciNet  Article  Google Scholar 

  8. 8

    Du Y., Ding R.: New proofs about the number of empty convex 4-gons and 5-gons in a planar point set. J. Appl. Math. Comput. 19(1–2), 93–104 (2005). doi:10.1007/BF02935790

    MATH  MathSciNet  Article  Google Scholar 

  9. 9

    Dujmović V., Eppstein D., Suderman M., Wood D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38, 194–212 (2007). doi:10.1016/j.comgeo.2006.09.002

    MATH  Google Scholar 

  10. 10

    Dumitrescu A.: Planar sets with few empty convex polygons. Studia Sci. Math. Hungar. 36(1–2), 93–109 (2000). doi:10.1556/SScMath.36.2000.1-2.9

    MATH  MathSciNet  Google Scholar 

  11. 11

    Eppstein, D.: Happy endings for flip graphs. J. Comput. Geom., 1(1), 3–28 (2010). http://jocg.org/index.php/jocg/article/view/21

    Google Scholar 

  12. 12

    Erdős P.: On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. 4(103), 99–108 (1975). doi:10.1007/BF02414146

    Google Scholar 

  13. 13

    Erdős P.: On some metric and combinatorial geometric problems. Discrete Math. 60, 147–153 (1986). doi:10.1016/0012-365X(86)90009-9

    MathSciNet  Article  Google Scholar 

  14. 14

    Erdős, P.: Some old and new problems in combinatorial geometry. In: Applications of Discrete Mathematics, pp. 32–37. SIAM (1988)

  15. 15

    Erdős P.: Problems and results on extremal problems in number theory, geometry, and combinatorics. Rostock. Math. Kolloq. 38, 6–14 (1989)

    Google Scholar 

  16. 16

    Erdős P., Szekeres G.: A combinatorial problem in geometry. Composito Math. 2, 464–470 (1935)

    Google Scholar 

  17. 17

    Füredi Z.: Maximal independent subsets in Steiner systems and in planar sets. SIAM J. Discrete Math. 4(2), 196–199 (1991). doi:10.1137/0404019

    MATH  MathSciNet  Article  Google Scholar 

  18. 18

    Gerken T.: Empty convex hexagons in planar point sets. Discrete Comput. Geom. 39(1–3), 239–272 (2008). doi:10.1007/s00454-007-9018-x

    MATH  MathSciNet  Article  Google Scholar 

  19. 19

    Graham R.L., Rothschild B.L., Spencer J.H.: Ramsey Theory. Wiley, London (2005)

    Google Scholar 

  20. 20

    Harborth H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33(5), 116–118 (1978)

    MATH  MathSciNet  Google Scholar 

  21. 21

    Horton J.D.: Sets with no empty convex 7-gons. Canad. Math. Bull. 26(4), 482–484 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22

    Hosono K.: On the existence of a convex point subset containing one triangle in the plane. Discrete Math. 305(1–3), 201–218 (2005). doi:10.1016/j.disc.2005.07.006

    MATH  MathSciNet  Article  Google Scholar 

  23. 23

    Kára J., Pór A., Wood D.R.: On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom. 34(3), 497–506 (2005). doi:10.1007/s00454-005-1177-z

    MATH  MathSciNet  Article  Google Scholar 

  24. 24

    Koshelev V.A.: The Erdős-Szekeres problem. Dokl. Akad. Nauk 415(6), 734–736 (2007)

    MathSciNet  Google Scholar 

  25. 25

    Kun G., Lippner G.: Large empty convex polygons in k-convex sets. Period. Math. Hungar. 46(1), 81–88 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  26. 26

    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, Berlin (2002). ISBN 0-387-95373-6

  27. 27

    Matoušek J.: Blocking visibility for points in general position. Discrete Comput. Geom. 42(2), 219–223 (2009). doi:10.1007/s00454-009-9185-z

    MATH  MathSciNet  Article  Google Scholar 

  28. 28

    Morris W.D., Valeriu S.: The Erdős–Szekeres problem on points in convex position—a survey. Bull. Am. Math. Soc. (N.S.) 37(4), 437–458 (2000)

    MATH  Article  Google Scholar 

  29. 29

    Nicolás C.M.: The empty hexagon theorem. Discrete Comput. Geom. 38(2), 389–397 (2007). doi:10.1007/s00454-007-1343-6

    MATH  MathSciNet  Article  Google Scholar 

  30. 30

    Nyklová H.: Almost empty polygons. Studia Sci. Math. Hungar. 40(3), 269–286 (2003). doi:10.1556/SScMath.40.2003.3.1

    MATH  MathSciNet  Google Scholar 

  31. 31

    Overmars M.: Finding sets of points without empty convex 6-gons. Discrete Comput. Geom. 29(1), 153–158 (2003). doi:10.1007/s00454-002-2829-x

    MATH  MathSciNet  Google Scholar 

  32. 32

    Pinchasi R., Radoičić R., Sharir M.: On empty convex polygons in a planar point set. J. Combin. Theory Ser. A 113(3), 385–419 (2006). doi:10.1016/j.jcta.2005.03.007

    MATH  MathSciNet  Article  Google Scholar 

  33. 33

    Pór, A., Wood, D.R.: On visibility and blockers. J. Comput. Geom. 1(1), 29–40 (2010). http://www.jocg.org/index.php/jocg/article/view/24

    Google Scholar 

  34. 34

    Rabinowitz S.: Consequences of the pentagon property. Geombinatorics 14, 208–220 (2005)

    Google Scholar 

  35. 35

    Tóth, G., Valtr, P.: The Erdős–Szekeres theorem: upper bounds and related results. In: Combinatorial and Computational Geometry. Math. Sci. Res. Inst. Publ., vol. 52, pp. 557–568. Cambridge Univ. Press, Cambridge (2005)

  36. 36

    Valtr P.: Convex independent sets and 7-holes in restricted planar point sets. Discrete Comput. Geom. 7(2), 135–152 (1992). doi:10.1007/BF02187831

    MATH  MathSciNet  Article  Google Scholar 

  37. 37

    Valtr P.: On the minimum number of empty polygons in planar point sets. Studia Sci. Math. Hungar. 30(1–2), 155–163 (1995)

    MATH  MathSciNet  Google Scholar 

  38. 38

    Valtr P.: A sufficient condition for the existence of large empty convex polygons. Discrete Comput. Geom. 28(4), 671–682 (2002). doi:10.1007/s00454-002-2898-x

    MATH  MathSciNet  Google Scholar 

  39. 39

    Valtr, P.: On empty hexagons. In: Surveys on Discrete and Computational Geometry. Contemp. Math. Amer. Math. Soc., vol. 453, pp. 433–441 (2008)

  40. 40

    Valtr P., Lippner G., Károlyi G.: Empty convex polygons in almost convex sets. Period. Math. Hungar. 55(2), 121–127 (2007). doi:10.1007/s10998-007-4121-z

    MATH  MathSciNet  Article  Google Scholar 

  41. 41

    Wu, L., Ding, R.: Reconfirmation of two results on disjoint empty convex polygons. In: Discrete Geometry, Combinatorics and Graph Theory. Lecture Notes in Comput. Sci., vol. 4381, pp. 216–220. Springer, Berlin (2007)

  42. 42

    Wu L., Ding R.: On the number of empty convex quadrilaterals of a finite set in the plane. Appl. Math. Lett. 21(9), 966–973 (2008). doi:10.1016/j.aml.2007.10.011

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to David R. Wood.

Additional information

Prosenjit Bose Research supported by NSERC. Ferran Hurtado partially supported by projects MTM2009-07242 and 2009SGR1040. David R. Wood supported by a QEII Research Fellow from the Australian Research Council.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Abel, Z., Ballinger, B., Bose, P. et al. Every Large Point Set contains Many Collinear Points or an Empty Pentagon. Graphs and Combinatorics 27, 47–60 (2011). https://doi.org/10.1007/s00373-010-0957-2

Download citation

Keywords

  • Erdős–Szekeres theorem
  • Happy end problem
  • Big line or big clique conjecture
  • Empty quadrilateral
  • Empty pentagon
  • Empty hexagon

Mathematics Subject Classification (2000)

  • 52C10 (Erdős problems and related topics of discrete geometry)
  • 05D10 (Ramsey theory)