On the Existence of Ordinary Triangles

2017/01/27 by Radoslav Fulek, Hossein Nassajian Mojarrad, Fulek, Radoslav +9
Computer Science · Mathematics · #52C30 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Mathematics and Applications

paper · doi:10.48550/arxiv.1701.08183

openalex publication_date 2017/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Motivated by a question of Erdős, and answering a question of de Zeeuw, 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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