Large vertex-flames in uncountable digraphs

The study of minimal subgraphs witnessing a connectivity property is an important field in graph theory. The foundation for large flames has been laid by Lov\'asz: Let $D=(V,E)$ be a finite digraph and let $r\in V$. 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. Lov\'asz proved that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $L$ sufficing the min-max criteria from Lov\'asz' theorem may instead also be captured by the following structural 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 $D$. The positive result for countably infinite digraphs based on this structural infinite generalisation were obtained by the second author. In this paper we extend this to digraphs of size $\aleph_1$ which requires significantly more complex techniques. Despite solving yet the smallest uncountable case, the complete understanding of the concept and potentially a proof for arbitrary cardinality still seems to be far.