Independent Hyperplanes in Oriented Paving Matroids

Abstract

In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank $3$, in this paper, we will use this result to prove by induction that an oriented paving matroid of rank $r\ge 3$ on $n$ elements, where $n\ge 5+r$, has at least $\frac{12}{13(r-1)}\left(\genfrac{}{}{0ex}{}{n}{r-2}\right)$ independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.