Edmonds' Branching Theorem in Digraphs without Forward-infinite Paths

Let $D$ be a finite digraph, and let $V_0,\dots,V_{k-1}$ be nonempty subsets of $V(D)$. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $\mathcal{B}_0,\dots, \mathcal{B}_{k-1}$ in $D$ such that the root set of $\mathcal{B}_i$ is $V_i\ (i=0,\dots,k-1)$ if and only if for all $\varnothing \neq X\subseteq V(D)$ the number of ingoing edges of $X$ is greater than or equal to the number of sets $V_i$ disjoint from $X$. As was shown by R. Aharoni and C. Thomassen in \cite{aharoni1989infinite}, this theorem does not remain true for infinite digraphs. Thomassen also proved that for the class of digraphs without backward-infinite paths, the above theorem of Edmonds remains true. Our main result is that for digraphs without forward-infinite paths, Edmonds' branching theorem remains true as well.