Countable Menger theorem with finitary matroid constraints on the ingoing edges

Abstract

We present a strengthening of the countable Menger theorem (edge version) of R. Aharoni. Let $D=(V,A)$ be a countable digraph with $s\ne t\in V$ and let $\mathcal{M}={⨁}_{v\in V}{\mathcal{M}}_v$ be a matroid on $A$ where ${\mathcal{M}}_v$ is a finitary matroid on the ingoing edges of $v$. We show that there is a system of edge-disjoint $s\to t$ paths $\mathcal{P}$ such that the united edge set of the paths is $\mathcal{M}$-independent, and there is a $C\subseteq A$ consists of one edge from each element of $\mathcal{P}$ for which ${\mathsf{span}}_{\mathcal{M}}(C)$ covers all the $s\to t$ paths in $D$.