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 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.