Unavoidable Minors of Matroids with Minimum Cocircuit Size Four

TL;DR

This paper generalizes classical graph minor results to matroids, showing that simple matroids with all cocircuits of size at least four necessarily contain one of nine specific minors, all of small rank and size.

Contribution

It extends known theorems from graphs and binary matroids to a broader class of simple matroids with cocircuit size constraints.

Findings

Identifies nine specific minors for such matroids.

All these minors have rank at most five and at most twelve elements.

Provides a unifying framework for minors in matroids with cocircuit size conditions.

Abstract

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which every cocircuit has size at least four has $F_7, M^*(K_{3,3}), M(K_5),$ or $ M(K_{2,2,2})$ as a minor. Generalizing these results, this paper proves that every simple matroid in which all cocircuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements.