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 Ω(|P|).
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- OntheExistenceofOrdinaryTriangles
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- 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 }