Hindrance from a wasteful common independent set

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.