On the packing/covering conjecture of infinite matroids

Abstract

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 \(({M_i}:i \in \Theta)\) 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 ∈ Θ) with \({\cup _{i \in \Theta}}{I_i} = E\) 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 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.