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

Authors:
Zachary Abel;Brad Ballinger;Prosenjit Bose;Sébastien Collette;Vida Dujmović;Ferran Hurtado;Scott Duke Kominers;Stefan Langerman;Attila Pór;David R. Wood
Affiliations:
Harvard University, Department of Mathematics, Cambridge, MA, USA;Humboldt State University, Department of Mathematics, California, USA;Carleton University, School of Computer Science, Ottawa, Canada;Université Libre de Bruxelles, Chargé de Recherches du F.R.S.-FNRS, Département d’Informatique, Brussels, Belgium;Carleton University, School of Computer Science, Ottawa, Canada;Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada II, Barcelona, Spain;Harvard University, and Harvard Business School, Department of Economics, Boston, MA, USA;Université Libre de Bruxelles, Maître de Recherches du F.R.S.-FNRS, Département d’Informatique, Brussels, Belgium;Western Kentucky University, Department of Mathematics, Bowling Green, Kentucky, USA;The University of Melbourne, Department of Mathematics and Statistics, Melbourne, Australia
Venue:
Graphs and Combinatorics
Year:
2011

Citing 0
Cited 3

On planar point sets with the pentagon property

Proceedings of the twenty-ninth annual symposium on Computational geometry
Unsolved problems in visibility graphs of points, segments, and polygons

ACM Computing Surveys (CSUR)
Flip distance between triangulations of a planar point set is APX-hard

Computational Geometry: Theory and Applications

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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].