Intersection of a partitional and a general infinite matroid

Abstract

Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $ I_M\sqcup I_N $ such that $ \mathsf{span}_M(I_M)\cup \mathsf{span}_N(I_N)=E $. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special case where one of the matroids is a direct sum of uniform matroids, i.e., is a partitional matroid. We show that if $ M $ is an arbitrary matroid and $ N $ is the direct sum of finitely many uniform matroids, then $ \{ M, N \} $ has the Intersection property.