Matroid fragility and relaxations of circuit hyperplanes

TL;DR

This paper explores the relationship between two key conjectures in matroid theory related to branch-width bounds and relaxations of circuit hyperplanes, contributing to the proof of Rota's Conjecture.

Contribution

It proves that the conjecture about relaxations of circuit hyperplanes implies the conjecture about bounded branch-width of fragile matroids.

Findings

The second conjecture implies the first.

Bounded branch-width depends only on the field and matroid properties.

Supports progress towards Rota's Conjecture.

Abstract

We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let $\mathbb F$ be a finite field. The first conjecture states that: the branch-width of any $\mathbb F$-representable $N$-fragile matroid is bounded by a function depending only upon $\mathbb F$ and $N$. The second conjecture states that: if a matroid $M_2$ is obtained from a matroid $M_1$ by relaxing a circuit-hyperplane and both $M_1$ and $M_2$ are $\mathbb F$-representable, then the branch-width of $M_1$ is bounded by a function depending only upon $\mathbb F$. Our main result is that the second conjecture implies the first.