On the packing/covering conjecture of infinite matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family (M_i:i ∈Θ) of matroids on the common edge set E is a system (S_i:i ∈Θ) of pairwise disjoint subsets of E where S i is panning in M i . Similarly, a covering is a system ( I i : i ∈ Θ) with ∪ _i ∈ΘI_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⊔E_c such that (M_i↾E_p:i ∈Θ) admits a packing and (M_i.E_c:i ∈Θ) admits a covering. We prove the case where E is countable and each M i is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.