Almost every matroid has an $M(K_4)$- or a $\mathcal{W}^3$-minor

TL;DR

This paper demonstrates that nearly all matroids include either the rank-3 whirl or the complete-graphic matroid as a minor, highlighting common structural features in matroid theory.

Contribution

It establishes that almost every matroid contains a specific minor, either $ ext{W}^3$ or $M(K_4)$, revealing prevalent structural minors in matroids.

Findings

Almost all matroids contain $ ext{W}^3$ or $M(K_4)$ as minors.

The presence of these minors is nearly universal in large matroids.

Structural minors like $ ext{W}^3$ and $M(K_4)$ are common in matroid classes.

Abstract

We show that almost every matroid contains the rank-3 whirl $\mathcal{W}^3$ or the complete-graphic matroid $M(K_4)$ as a minor.