Intersection of a partitional and a general infinite matroid

TL;DR

This paper proves that for an arbitrary matroid and a finite direct sum of uniform matroids, the pair satisfies the Intersection property, advancing understanding of the Matroid Intersection Conjecture.

Contribution

It establishes the Intersection property for pairs where one matroid is arbitrary and the other is a finite sum of uniform matroids, a case related to the Matroid Intersection Conjecture.

Findings

Proves the Intersection property for specific matroid pairs.

Extends known cases of the Matroid Intersection Conjecture.

Provides new insights into infinite matroid intersections.

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.