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Interesting Esoterica

On the Existence of Ordinary Triangles

Article by Radoslav Fulek and Hossein Nassajian Mojarrad and Márton Naszódi and József Solymosi and Sebastian U. Stich and May Szedlák
  • 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 Ω(|P|).

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key
OntheExistenceofOrdinaryTriangles
type
article
date_added
2017-02-06
date_published
2017-06-24

BibTeX entry

@article{OntheExistenceofOrdinaryTriangles,
	key = {OntheExistenceofOrdinaryTriangles},
	type = {article},
	title = {On the Existence of Ordinary Triangles},
	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},
	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|){\$}.},
	comment = {},
	date_added = {2017-02-06},
	date_published = {2017-06-24},
	urls = {http://arxiv.org/abs/1701.08183v1,http://arxiv.org/pdf/1701.08183v1},
	collections = {Geometry,Fun maths facts},
	url = {http://arxiv.org/abs/1701.08183v1 http://arxiv.org/pdf/1701.08183v1},
	urldate = {2017-02-06},
	archivePrefix = {arXiv},
	eprint = {1701.08183},
	primaryClass = {math.CO},
	year = 2017
}