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Every Large Point Set contains Many Collinear Points or an Empty Pentagon |
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| Authors: | Zachary Abel Brad Ballinger Prosenjit Bose S??bastien Collette Vida Dujmovi? Ferran Hurtado Scott Duke Kominers Stefan Langerman Attila P??r David R Wood |
| Institution: | 1. Department of Mathematics, Harvard University, Cambridge, MA, USA 2. Department of Mathematics, Humboldt State University, California, USA 3. School of Computer Science, Carleton University, Ottawa, Canada 4. Charg?? de Recherches du F.R.S.-FNRS, D??partement d??Informatique, Universit?? Libre de Bruxelles, Brussels, Belgium 5. Departament de Matem??tica Aplicada II, Universitat Polit??cnica de Catalunya, Barcelona, Spain 6. Department of Economics, Harvard University, and Harvard Business School, Boston, MA, USA 7. Ma?tre de Recherches du F.R.S.-FNRS, D??partement d??Informatique, Universit?? Libre de Bruxelles, Brussels, Belgium 8. Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky, USA 9. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia |
| 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]. |
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