  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.
