Hindrance from a wasteful common independent set

TL;DR

This paper advances the understanding of the Matroid Intersection Conjecture by introducing a new approach inspired by König's theorem, showing that certain 'wasteful' common independent sets imply the existence of hindrances.

Contribution

It presents a matroidal generalization of the vertex approach from König's theorem and links wasteful independent sets to hindrances, advancing the conjecture.

Findings

Identifies a new matroidal approach inspired by König's theorem.

Shows wasteful common independent sets imply hindrances.

Provides a breakthrough towards the Matroid Intersection Conjecture.

Abstract

For (potentially infinite) matroids $ M $ and $ N $, an $ (M,N) $-hindrance is a set $ H$ that is independent but not spanning in $ N.\mathsf{span}_M(H) $. This concept was introduced by Aharoni and Ziv in the very first paper investigating Nash-Williams' Matroid Intersection Conjecture. They proved that the conjecture is equivalent to the statement that the non-existence of hindrances implies the existence of an $ M $-independent spanning set of $ N $. In this paper we present a breakthrough towards the Matroid Intersection Conjecture. Namely, we found a matroidal generalization of the `popular vertex' approach applied in the proof of the infinite version of K\"onig's theorem. The main result of this paper is an application of this new approach to show that if $ M $ and $ N $ admit a common independent set $ I $ that is ``wasteful'' in the sense that $ r(M/I)<r(N/I) $, then there exists an $ (M,N) $-hindrance.