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 . 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 |).2010 Mathematics Subject Classification. 52C30.

A better bound for ordinary triangles

“…The proof for the first case follows that of Fulek et al [4]. Our main contribution is finding a better argument for the second case.…”

“…With an interest in studying ordinary conics [2], de Zeeuw asked whether there exists an integer c such that every set P of points not contained in the union of two lines contains a c-ordinary triangle, that is, three non-collinear points of P , where each line spanned by the points is c-ordinary. Fulek, Nassajian Mojarrad, Naszódi, Solymosi, Stich, and Szedlák [4] answered in the affirmative for n sufficiently large, and showed one may take c = 12000. We improve this to c = 11.…”

“…Erdős [3] was the first to consider conditions under which one can find a 2-ordinary triangle in a set of points, that is to say three non-collinear points such that each spanned line is 2-ordinary. While it is clear that the point set must not be contained in the union of two lines, each containing at least three points, the following example illustrates that this is not sufficient [4]: Given a set of points P and a line l not spanned by these points, add to P the points of intersection between l and each of the lines spanned by P , to get a new set of points P ′ . Any 2-ordinary triangle in P ′ must contain at least two points from the original point set P , however, each line spanned by P contains a third point on l. Therefore, there are no 2-ordinary triangles in P ′ .…”

Spanned lines and Langer's inequality

“…Fulek et al [10] proved that there exists a c > 0 such that if a finite point set in R 2 is not covered by two lines, then there are three points in the set such that all three lines spanned by them contain at most c points (they call this a c-ordinary triangle). Their proof gave c = 12000.…”

“…Compare this with the Szemerédi-Trotter theorem [23,24,25] in R 2 , which implies that the number of lines with at least k points is at most cn 2 /k 3 for some constant c (when k < √ n). This c is fairly large ( [10] states c = 125), so (9) does better for small k (specifically k ≤ 58).…”