Projective geometries in exponentially dense matroids. I

TL;DR

This paper characterizes the maximum density of certain minor-closed classes of matroids, showing a dichotomy based on the inclusion of all rank-$(a+1)$ uniform matroids and the presence of GF(q)-representable matroids.

Contribution

It establishes a polynomial density bound for matroids in minor-closed classes, distinguishing cases based on uniform matroids and GF(q)-representability.

Findings

Density bounds are polynomial in rank for classes excluding all rank-$(a+1)$ uniform matroids.

If GF(q)-representable matroids are included, density bounds grow exponentially with rank.

The results determine the maximum density up to polynomial factors for these classes.

Abstract

We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by at most $r^n$ sets of rank at most $a$, or $\cM$ contains the $\GF(q)$-representable matroids for some prime power $q$, and every rank-$r$ matroid in $\cM$ can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in $\cM$ up to a polynomial factor.