Minors of asymptotically almost all sparse paving matroids

TL;DR

This paper demonstrates that nearly all sparse paving matroids contain certain minors, revealing structural properties common to almost all such matroids through counting arguments.

Contribution

It establishes that asymptotically almost all sparse paving matroids contain minors from specific classes, extending to larger classes under fixed rank restrictions.

Findings

Nearly all sparse paving matroids contain specific minors.

The result applies to larger classes of minors for fixed rank.

Counting arguments underpin the proof.

Abstract

We use counting arguments to show that asymptotically almost all sparse paving matroids contain an $H$-minor, where $H$ falls into one of several simple classes of matroids. Furthermore the result holds for all $H$ in a larger class of matroids, if we restrict to asymptotically almost all sparse paving matroids of fixed rank $r$ (where $r$ is necessarily no smaller than the rank of $H$).