Florian Gut and Attila Joó, Large vertex-flames in uncountable digraphs

The local connectivity  $ \kappa_D(r,v) $ from $ r $ to $ v $ is defined to be the maximal number of internally disjoint $r\rightarrow v $ paths in $ D $. A spanning subdigraph $ L $ of $ D $ with $  \kappa_L(r,v)=\kappa_D(r,v) $ for every $ v\in V-r $ must have at least $ \sum_{v\in V-r}\kappa_D(r,v) $ edges. It was shown by Lovász that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $ L $ can be captured by the following characterization: For every $ v\in V-r $ there is a system $ \mathcal{P}_v $ of internally disjoint $ r\rightarrow v $ paths in $ L $ covering all the ingoing edges of $ v $ in $ L $ such that one can choose from  each $ P\in \mathcal{P}_v $ either an edge or an internal vertex in such a way that the resulting set meets every $ r\rightarrow v $ path of $ D $. We prove that every digraph of size at most $ \aleph_1 $  admits such a spanning subdigraph $ L $. The question if this remains true for larger digraphs remains open.