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


We prove the following generalised empty pentagon theorem for every integer  ≥ 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 Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].

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Corresponding author

Correspondence to David R. Wood.

Additional information

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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Abel, Z., Ballinger, B., Bose, P. et al. Every Large Point Set contains Many Collinear Points or an Empty Pentagon. Graphs and Combinatorics 27, 47–60 (2011). https://doi.org/10.1007/s00373-010-0957-2

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  • Erdős–Szekeres theorem
  • Happy end problem
  • Big line or big clique conjecture
  • Empty quadrilateral
  • Empty pentagon
  • Empty hexagon

Mathematics Subject Classification (2000)

  • 52C10 (Erdős problems and related topics of discrete geometry)
  • 05D10 (Ramsey theory)