  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.
