An Application of Rado's Theorem to Disjoint Matchings in Bipartite Graphs

For the terminology and notation we follow Bondy and Murty [1] in graph theory and Welsh [3] in matroid theory. All sets considered are finite. Let G be a bipartite graph with bipartition (X, Y) and let V and E denote, respectively, the vertex and edge sets of G. A matching M in G is called a matching of X into Y if every vertex in X is M-saturated. If S is any subset of V, we shall write <5(S) to denote the set of edges incident with S; we simplify 5({y}) to 5(y). Lebensold [2] proved the following result.