Menger's theorem for countable graphs

For a finite graph G= (I’, E) Menger’s Theorem [6] states the following: if A, B c V then the minimal size of an A -B separating set of vertices (i.e., a set whose removal disconnects A from B), equals the maximal size of a set of disjoint A -B paths. This version of the theorem remains true, and quite easy to prove, also in the infinite case. To see this, take 9 to be a maximal (with respect to containment) set of A -B paths (such exists by Zorn’s Lemma.) Clearly, S = u ( V(P) : P E 9’} is an A B separating set, and hence the size of any set of disjoint A B paths cannot exceed 1 SI . If B is finite this implies that there exists a finite family 9 of disjoint A -B paths of maximal size. In this case it can be shown by an alternating paths method (see, e.g., the proof of the max-flow min-cut theorem [4]), that there exists an A B separating set of vertices consisting of the choice of one vertex from each path in 9, proving the theorem. If 9 is infinite then 1 SI = 19 1, and again the desired result follows. But this version does not capture the strength of the finite theorem-in the proof the separating set chosen contains “too many” vertices. Erdtjs (see, e.g., [3, 71) suggested the following stronger version, which in the finite case is clearly equivalent to the version stated above: