# On the Existence of Ordinary Triangles

## TL;DR

This paper proves that any finite point set in the plane, not confined to two lines, contains a constant proportion of triangles where each side has at most a fixed number of points, answering a question related to Erdős.

## Contribution

It establishes the existence of a universal constant c ensuring the presence of c-ordinary triangles in any non-degenerate point set, and shows the linear lower bound on their quantity.

## Key findings

- Existence of a universal constant c for c-ordinary triangles.
- Any such point set contains linearly many c-ordinary triangles.
- Addresses a question posed by Erdős and de Zeeuw.

## Abstract

Let $P$ be a finite point set in the plane. A \emph{$c$-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 Erd\H{o}s, 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 $\Omega(|P|)$.