1. Introduction

Sylvester [1893] raised the following problem: Given finitely many noncollinear points in the plane, do they always span a simple line (that is, a line that contains precisely two of the points)? The answer is yes, as was first shown by Gallai [1944].

By duality, the former question is equivalent to the question: Given finitely many straight lines in the plane, not all passing through the same point, do they always determine a simple intersection point (a point that lies on precisely two of the lines)?

A natural generalization is to find a lower bound on the number of simple lines (or simple points, in the dual version). The dual version of this question can be generalized to pseudolines. The best lower bound [Csima and Sawyer 1993] states that an arrangement of n pseudolines in the plane determines at least 6n/13 simple points. The conjecture [Borwein and Moser 1990] is that there are at least n/2 simple points for n ≠ 7,13. For the history of Sylvester's problem, with its many proofs and generalizations, see [Borwein and Moser 1990; Nilakantan 2005].