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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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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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