Milner-Shelah transversal theorem revisited

The Milner-Shelah transversal theorem states that if G=(A,B,E) is a (possibly infinite) bipartite graph such that there is no isolated vertex in A, and for every edge {a,b} in E with a in A and b in B, d_G(a)>= d_G(b), then G has a matching covering A. We recall the Marriage theorem by Aharoni, Nash-Williams and Shelah and provide a new proof of the Milner-Shelah theorem.