# On the Existence of Ordinary Triangles

• Published in 2017
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Let $P$ be a finite point set in the plane. A $c$-ordinary triangle in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of $P$. We prove that there exists a constant $c>0$ such that $P$ contains a $c$-ordinary triangle, provided that $P$ is not contained in the union of two lines. Furthermore, the number of $c$-ordinary triangles in $P$ is $\Omega(|P|)$.

### BibTeX entry

@article{OntheExistenceofOrdinaryTriangles,
title = {On the Existence of Ordinary Triangles},
abstract = {Let {\$}P{\$} be a finite point set in the plane. A {\$}c{\$}-ordinary triangle in {\$}P{\$} is
a subset of {\$}P{\$} consisting of three non-collinear points such that each of the
three lines determined by the three points contains at most {\$}c{\$} points of {\$}P{\$}.
We prove that there exists a constant {\$}c>0{\$} such that {\$}P{\$} contains a
{\$}c{\$}-ordinary triangle, provided that {\$}P{\$} is not contained in the union of two
lines. Furthermore, the number of {\$}c{\$}-ordinary triangles in {\$}P{\$} is
{\$}\Omega(|P|){\$}.},
url = {http://arxiv.org/abs/1701.08183v1 http://arxiv.org/pdf/1701.08183v1},
author = {Radoslav Fulek and Hossein Nassajian Mojarrad and M{\'{a}}rton Nasz{\'{o}}di and J{\'{o}}zsef Solymosi and Sebastian U. Stich and May Szedl{\'{a}}k},
comment = {},
urldate = {2017-02-06},
archivePrefix = {arXiv},
eprint = {1701.08183},
primaryClass = {math.CO},
year = 2017,
collections = {Geometry,Fun maths facts}
}