On a Problem of Groübaum

Abstract

Pn will denote a set of n points in the plane. A well known theorem of Gallai- Sylvester (see e.g. [4]) states that if the points of Pn do not all lie on a line then they always determine an ordinary line, i.e. a line which goes through precisely two of the points of Pn.Using this theorem I proved that if the points do not all lie on a line, they determine at least n lines. I conjectured that if n>n0 and no n—1 points of Pn are on a line, they determine at least 2n-4 lines. This conjecture was proved by Kelly and Moser [3], who, in fact, proved the following more general result: Let Pn be such that at most n—k of its points are collinear.