For (potentially infinite) matroids M and N, an (M,N)-hindrance is a set H that is independent but not spanning in N.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.

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In this talk we present a recent 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 König's theorem. We sketch the proof of the following statement: If matroids 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.