Every Large Point Set contains Many Collinear Points or an Empty Pentagon
نویسندگان
چکیده
منابع مشابه
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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An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328` points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the (` − 1) × (` − 1) grid contains no empty pentagon and no ` collinear points. ...
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ژورنال
عنوان ژورنال: Graphs and Combinatorics
سال: 2010
ISSN: 0911-0119,1435-5914
DOI: 10.1007/s00373-010-0957-2