Every Large Point Set contains Many Collinear Points or an Empty Pentagon
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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].
KeywordsErdős–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)
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