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 Email author Original Paper First Online: 07 October 2010 Received: 27 April 2009 Revised: 09 June 2010 DOI :
10.1007/s00373-010-0957-2
Cite this article as: Abel, Z., Ballinger, B., Bose, P. et al. Graphs and Combinatorics (2011) 27: 47. doi:10.1007/s00373-010-0957-2 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 Prosenjit Bose Research supported by NSERC. Ferran Hurtado partially supported by projects MTM2009-07242 and 2009SGR1040. David R. Wood supported by a QEII Research Fellow from the Australian Research Council.
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