Dense PG(n-1,2)-free binary matroids

TL;DR

This paper proves that large binary matroids avoiding a projective geometry of dimension n-1 contain a specific flat with no triangles, revealing structural properties of such matroids.

Contribution

It establishes a new extremal bound for binary matroids excluding a projective geometry, identifying the existence of a triangle-free flat under certain size conditions.

Findings

Large PG(n-1,2)-free binary matroids have a triangle-free flat of corank n-2.

The size threshold for the matroid guarantees the existence of the flat.

The result extends understanding of the structure of binary matroids avoiding certain projective geometries.

Abstract

For each integer $n \geq 2$, we prove that, if $M$ is a simple rank-$r$ $PG(n-1,2)$-free binary matroid with $|M|>\left(1-\frac{3}{2^n}\right)2^r$, then there is a triangle-free corank-$(n-2)$ flat of $M$.