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

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

2009-04-01

arxiv.org/abs/0904.0262v2

Graphs and Combinatorics 27(1), (2011), 47-60

Mathematics

Combinatorics

Scientific paper

10.1007/s00373-010-0957-2

We prove the following generalised empty pentagon theorem: for every integer
$\ell \geq 2$, every sufficiently large set of points in the plane contains
$\ell$ collinear points or an empty pentagon. As an application, we settle the
next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and
Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].

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