Projective geometries in exponentially dense matroids. II

TL;DR

This paper characterizes the structure and density of minor-closed classes of matroids, showing that such classes either have polynomially bounded coverings or contain dense GF(q)-representable matroids, with implications for their maximum density.

Contribution

It establishes a dichotomy for minor-closed matroid classes based on their density and structural properties, extending projective geometry concepts to dense matroids.

Findings

Classes either have polynomial covering bounds or contain dense GF(q)-representable matroids.

Maximum density of matroids in these classes is determined up to a constant factor.

Provides a structural characterization linking density and representability.

Abstract

We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant factor.