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.

堆积/覆盖猜想是由 Bowler 和 Carmesin 在 Edmonds 和 Fulkerson 的拟阵划分定理的推动下提出的。公共边集E上的拟阵族\(({M_i}:i \in \Theta)\)的包装是成对不相交子集的系统\(({S_i}:i \in \Theta)\) E的其中S _i正在M _i中平移。类似地，覆盖是一个具有\({\cup _{i \in \Theta}}{I_i} = E\) 的系统 ( I _i : i ∈ θ) ，其中I _i独立于M _i。该猜想指出，对于E上的每个拟阵族，都有一个分区\(E = {E_p} \sqcup {E_c}\)使得\(({M_i}\upharpoonright{E_p}:i \in \Theta)\)承认包装并且\(({M_i}.{E_c}:i \in \Theta)\)承认覆盖。我们证明E是可数的，并且每个M _i是有限的或共有限的。为此，我们通过证明该猜想对于仅具有有限和共有限分量的可数拟阵成立，给出了 Ghaderi 的奇异拟阵交集定理和 Nash-Williams 拟阵交集猜想的可数情况的共同概括。