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 |
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01.01.2011
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| 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]. |
|---|---|
| AbstractList | 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]. We prove the following generalised empty pentagon theorem for every integer greater than or equal to 2, every sufficiently large set of points in the plane contains collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kara, Por, and Wood [Discrete Comput. Geom. 34(3):497-506, 2005]. 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]. We prove the following generalised empty pentagon theorem for every integer [greater than or equal to] 2, every sufficiently large set of points in the plane contains collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kara, Por, and Wood [Discrete Comput. Geom. 34(3):497-506, 2005]. [PUBLICATION ABSTRACT] |
| Author | Bose, Prosenjit Pór, Attila Kominers, Scott Duke Ballinger, Brad Dujmović, Vida Langerman, Stefan Wood, David R. Abel, Zachary Collette, Sébastien Hurtado, Ferran |
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| Cites_doi | 10.1090/S0273-0979-00-00877-6 10.1016/j.jcta.2005.03.007 10.1016/j.disc.2005.07.006 10.1090/conm/453/08809 10.1007/BF02187831 10.1016/j.aml.2007.10.011 10.1201/9780203911211.ch12 10.1007/BF02414146 10.1137/0404019 10.1007/s00454-009-9185-z 10.1007/3-540-47738-1_7 10.1007/s10998-004-0518-7 10.1007/BF01840404 10.1007/BF02935790 10.1023/A:1025757808886 10.1007/s00454-002-2829-x 10.1007/s10998-007-4121-z 10.1007/s00454-005-1177-z 10.4153/CMB-1983-077-8 10.1007/978-3-540-70666-3_23 10.1007/978-1-4613-0039-7 10.1016/j.comgeo.2006.09.002 10.1007/s00454-007-9018-x 10.1016/0012-365X(86)90009-9 10.1007/s00454-007-1343-6 10.1007/s00454-002-2898-x |
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| Contributor | Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II |
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| Keywords | 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 |
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Dobkin (957_CR7) 1990; 5 P. Valtr (957_CR40) 2007; 55 V. Dujmović (957_CR9) 2007; 38 Z. Füredi (957_CR17) 1991; 4 957_CR26 P. Erdős (957_CR12) 1975; 4 957_CR2 I. Bárány (957_CR3) 2004; 41 957_CR1 K. Hosono (957_CR22) 2005; 305 H. Nyklová (957_CR30) 2003; 40 P. Valtr (957_CR36) 1992; 7 A. Dumitrescu (957_CR10) 2000; 36 P. Erdős (957_CR16) 1935; 2 P. Erdős (957_CR15) 1989; 38 J.D. Horton (957_CR21) 1983; 26 957_CR5 G. Kun (957_CR25) 2003; 46 P. Braß (957_CR6) 2005 H. Harborth (957_CR20) 1978; 33 C.M. Nicolás (957_CR29) 2007; 38 P. Valtr (957_CR38) 2002; 28 957_CR11 957_CR33 L. Wu (957_CR42) 2008; 21 957_CR14 T. Gerken (957_CR18) 2008; 39 957_CR35 Y. Du (957_CR8) 2005; 19 957_CR39 M. Overmars (957_CR31) 2003; 29 P. Erdős (957_CR13) 1986; 60 W.D. Morris (957_CR28) 2000; 37 P. Valtr (957_CR37) 1995; 30 J. Kára (957_CR23) 2005; 34 |
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| Snippet | 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... We prove the following generalised empty pentagon theorem for every integer [greater than or equal to] 2, every sufficiently large set of points in the plane... We prove the following generalised empty pentagon theorem for every integer greater than or equal to 2, every sufficiently large set of points in the plane... 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... |
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| SubjectTerms | 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 |
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| Title | Every Large Point Set contains Many Collinear Points or an Empty Pentagon |
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