On the Packing/Covering Conjecture of Infinite Matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family $(M_i: i\in\Theta)$ of matroids on the common edge set $E$ is a system $(S_i: i\in\Theta )$ of pairwise disjoint subsets of $E$ where $S_i$ is panning in $M_i$. Similarly, a covering is a system $(I_i: i\in\Theta )$ with $\bigcup_{i\in\Theta} I_i=E$ where $I_i$ is independent in $M_i$. The conjecture states that for every matroid family on $E$ there is a partition $E=E_p \sqcup E_c$ such that $(M_i \upharpoonright E_p: i\in \Theta)$ admits a packing and $(M_i. E_c: i\in \Theta)$ admits a covering. We prove the special case where $E$ is countable and each $M_i$ is either finitary or cofinitary. The connection between packing/covering and matroid intersection problems discovered by Bowler and Carmesin can be established for every well-behaved matroid class. This makes possible to approach the problem from the direction of matroid intersection. We show that the generalized version of Nash-Williams' Matroid Intersection Conjecture holds for countable matroids having only finitary and cofinitary components.