A better bound for ordinary triangles

Abstract

Let P be a finite set of points in the plane. A c-ordinary triangle is a set of three non-collinear points of P such that each line spanned by the points contains at most c points of P . We show that if P is not contained in the union of two lines and |P | is sufficiently large, then it contains an 11-ordinary triangle. This improves upon a result of Fulek et al., who showed one may take c = 12000.