Graphs and Combinatorics

, Volume 27, Issue 1, pp 47–60 | Cite as

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

  • Zachary Abel
  • Brad Ballinger
  • Prosenjit Bose
  • Sébastien Collette
  • Vida Dujmović
  • Ferran Hurtado
  • Scott Duke Kominers
  • Stefan Langerman
  • Attila Pór
  • David R. Wood
Original Paper
First Online:
Received:
Revised:

Abstract

We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ 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].

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) 
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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)Google Scholar
  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)MATHMathSciNetGoogle 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 MATHMathSciNetCrossRefGoogle 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)Google Scholar
  6. 6.
    Braß P., Moser W.O.J., Pach J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)MATHGoogle 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 MATHMathSciNetCrossRefGoogle 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 MATHMathSciNetCrossRefGoogle 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 MATHGoogle 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 MATHMathSciNetGoogle 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 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Erdős, P.: Some old and new problems in combinatorial geometry. In: Applications of Discrete Mathematics, pp. 32–37. SIAM (1988)Google Scholar
  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 MATHMathSciNetCrossRefGoogle 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 MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetGoogle Scholar
  21. 21.
    Horton J.D.: Sets with no empty convex 7-gons. Canad. Math. Bull. 26(4), 482–484 (1983)MATHMathSciNetCrossRefGoogle 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 MATHMathSciNetCrossRefGoogle 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 MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Koshelev V.A.: The Erdős-Szekeres problem. Dokl. Akad. Nauk 415(6), 734–736 (2007)MathSciNetGoogle Scholar
  25. 25.
    Kun G., Lippner G.: Large empty convex polygons in k-convex sets. Period. Math. Hungar. 46(1), 81–88 (2003)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, Berlin (2002). ISBN 0-387-95373-6Google Scholar
  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 MATHMathSciNetCrossRefGoogle 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)MATHCrossRefGoogle 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 MATHMathSciNetCrossRefGoogle 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 MATHMathSciNetGoogle 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 MATHMathSciNetGoogle 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 MATHMathSciNetCrossRefGoogle 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)Google Scholar
  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 MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetGoogle 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 MATHMathSciNetGoogle 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)Google Scholar
  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 MATHMathSciNetCrossRefGoogle 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)Google Scholar
  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 MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Zachary Abel
    • 1
  • Brad Ballinger
    • 2
  • Prosenjit Bose
    • 3
  • Sébastien Collette
    • 4
  • Vida Dujmović
    • 3
  • Ferran Hurtado
    • 5
  • Scott Duke Kominers
    • 6
  • Stefan Langerman
    • 7
  • Attila Pór
    • 8
  • David R. Wood
    • 9
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsHumboldt State UniversityCaliforniaUSA
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada
  4. 4.Chargé de Recherches du F.R.S.-FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  5. 5.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  6. 6.Department of EconomicsHarvard University, and Harvard Business SchoolBostonUSA
  7. 7.Maître de Recherches du F.R.S.-FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  8. 8.Department of MathematicsWestern Kentucky UniversityKentuckyUSA
  9. 9.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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