Independent Hyperplanes in Oriented Paving Matroids

TL;DR

This paper extends a known result about pseudoline arrangements to prove a new lower bound on the number of independent hyperplanes in oriented paving matroids of rank at least 3, providing a necessary condition for orientability.

Contribution

It introduces a novel lower bound on independent hyperplanes in oriented paving matroids, generalizing previous pseudoline intersection results to higher ranks.

Findings

Establishes a lower bound of (12/13(r-1)) * C(n, r-2) on independent hyperplanes

Provides a necessary condition for paving matroids to be orientable

Extends topological pseudoline intersection results to higher-dimensional 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 $n \geq 5+ r$, has at least $\frac{12}{13(r-1)} \binom{n}{r-2}$ independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.