The Highly Connected Matroids in Minor-closed Classes

TL;DR

This paper provides an exact structural characterization of highly connected matroids within minor-closed classes over finite fields, offering a foundation for future conjectures and research directions.

Contribution

It introduces a precise structural description for highly connected matroids in minor-closed classes over finite fields, advancing understanding in matroid theory.

Findings

Structural characterization of highly connected matroids

Formulation of new conjectures based on the characterization

Potential pathways for proving related conjectures

Abstract

For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.