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

doi:10.1007/s00373-010-0957-2

Graphs and Combinatorics

January 2011, Volume 27, Issue 1, pp 47–60

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

Authors
Authors and affiliations

Zachary Abel
Brad Ballinger
Prosenjit Bose
Sébastien Collette
Vida Dujmović
Ferran Hurtado
Scott Duke Kominers
Stefan Langerman
Attila Pór
David R. WoodEmail author

Original Paper

First Online:: 07 October 2010

Received:: 27 April 2009
Revised:: 09 June 2010

4 Citations
142 Downloads

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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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
Email author

1.Department of MathematicsHarvard UniversityCambridgeUSA
2.Department of MathematicsHumboldt State UniversityCaliforniaUSA
3.School of Computer ScienceCarleton UniversityOttawaCanada
4.Chargé de Recherches du F.R.S.-FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
5.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
6.Department of EconomicsHarvard University, and Harvard Business SchoolBostonUSA
7.Maître de Recherches du F.R.S.-FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
8.Department of MathematicsWestern Kentucky UniversityKentuckyUSA
9.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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

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