# Countable Menger theorem with finitary matroid constraints on the ingoing edges

## TL;DR

This paper strengthens the countable Menger theorem by incorporating finitary matroid constraints on ingoing edges, ensuring the existence of edge-disjoint paths with independent edge sets and a spanning subset covering all paths.

## Contribution

It introduces a new matroid-based framework for the countable Menger theorem, accommodating finitary matroid constraints on ingoing edges.

## Key findings

- Existence of edge-disjoint s→t paths with matroid independence
- Construction of a spanning set covering all s→t paths
- Extension of Menger's theorem to matroid-constrained digraphs

## 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\neq t\in V $ and let $\mathcal{M}=\bigoplus_{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 \rightarrow 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\rightarrow t $ paths in $ D $.