On a possible extension of Hall's theorem to bipartite hypergraphs

AbstractIn [1] an extension of Hall's theorem was conjectured for n-partite n-graphs and its fractional version was proved. It seems that the conjecture can be strengthened to apply any bipartite hypergraph (i.e. a hypergraph with a distinguished set of vertices A such that |e∩A| = 1 for every edge e). We prove the strengthened conjecture in the case that |A|⩽4 and also give a proof for its fractional version