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 $$({M_i}:i \in \Theta)$$ ( M i : i ∈ Θ ) of matroids on the common edge set E is a system $$({S_i}:i \in \Theta)$$ ( S i : i ∈ Θ ) of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with $${\cup _{i \in \Theta}}{I_i} = E$$ ∪ i ∈ Θ I i = E where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition $$E = {E_p} \sqcup {E_c}$$ E = E p ⊔ E c such that $$({M_i}\upharpoonright{E_p}:i \in \Theta)$$ ( M i ↾ E p : i ∈ Θ ) admits a packing and $$({M_i}.{E_c}:i \in \Theta)$$ ( M i . E c : i ∈ Θ ) admits a covering. We prove the case where E is countable and each Mi 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.