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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Veröffentlicht in:	Graphs and combinatorics Jg. 27; H. 1; S. 47 - 60
Hauptverfasser:	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 Verlag
Sprache:	Englisch
Veröffentlicht:	Japan Springer Japan 01.01.2011 Springer Nature B.V
Schlagworte:	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
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Zusammenfassung:	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].
Bibliographie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23
ISSN:	0911-0119 1435-5914
DOI:	10.1007/s00373-010-0957-2