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

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...

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Bibliographic Details
Published in:Graphs and combinatorics Vol. 27; no. 1; pp. 47 - 60
Main Authors: Abel, Zachary, Ballinger, Brad, Bose, Prosenjit, Collette, Sébastien, Dujmović, Vida, Hurtado, Ferran, Kominers, Scott Duke, Langerman, Stefan, Pór, Attila, Wood, David R.
Format: Journal Article Publication
Language:English
Published: Japan Springer Japan 01.01.2011
Springer Nature B.V
Subjects:
05 Combinatorics
05D Extremal combinatorics
52 Convex and discrete geometry
52C Discrete geometry
Classificació AMS
Combinatorial analysis
Combinatorics
Convex geometry
Discrete geometry
Engineering Design
Geometria
Geometria convexa
Geometria convexa i discreta
Geometria discreta
Geometry
Graphs
Hexagons
Integers
Matemàtiques i estadística
Mathematics
Mathematics and Statistics
Original Paper
Pentagon
Planes
Ramsey theory
Ramsey, Teoria de
Set theory
Theorems
Wood
Àrees temàtiques de la UPC
Happy end problem
05D10 (Ramsey theory)
Erdős–Szekeres theorem
52C10 (Erdős problems and related topics of discrete geometry)
Big line or big clique conjecture
Empty quadrilateral
Empty hexagon
Empty pentagon
ISSN:0911-0119, 1435-5914
Online Access:Get full text
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Summary: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].
Bibliography:SourceType-Scholarly Journals-1
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-010-0957-2