The directions determined by n points in the plane: a matroidal generalization

AbstractP.R. Scott posed the problem of determining the minimum number of directions determined by n points which are not all collinear in the plane. We consider a generalization of this problem for oriented matroids. We prove the following theorem: Let M denote an oriented matroid of rank 3. Suppose M has a modular line L, such that the n points of M not in L are not all collinear. Then L has at least 12(n+3) points