Menger's Theorem and Matroids

Let G be a finite directed graph with X, Y disjoint subsets of the nodes of G. Menger's theorem [6] asserts that the maximum cardinal number of a set 0 of pairwise node disjoint paths from X to Y equals the minimum cardinal number of a set which separates X from Y. A special case of Menger's theorem occurs when all the edges of G have initial node in X and terminal node in Y (the bipartite situation). The resulting special case is the well-known Konig duality theorem [5]. In [1,2 and 3] generalizations of Konig's theorem were obtained by assuming that X and Y had matroids defined on them and then requiring that the set of initial (resp. terminal) nodes of 0 be independent in the matroid on X (resp. Y). The case where Y only had a matroid defined on it had already been handled by Rado [8]. Our purpose in this note is to raise Menger's theorem to the same level of generality.