Menger's theorem for infinite graphs

As a first-year undergraduate in 1931 in König's graph theory course, Paul Erdös proved an infinite generalization of Menger's theorem, which only appeared in 1936 at the end of König's classic book on graph theory [2]. The result was this:

Given any two disjoint sets \( A\) and \( B\) of vertices in any graph, the minimum cardinality of an \( A\)-\( B\) separating set of vertices is equal to the maximum number of vertex-disjoint \( A\)-\( B\) paths.

As observed by Aharoni [1] Paul's short proof suggests that this was not quite the ``right" generalization of Menger's theorem, something which Paul himself corrected some years later with the following:

Problem

For any two sets \(A\) and \(B\) of vertices in any graph, there exists a set \(\mathcal F\) of disjoint \(A\)- -\(B\) paths and an \(A\)- -\(B\) separating set \(S\) of vertices, such that \(S\) consists of exactly one vertex from each path in \(\mathcal F\).

While this conjecture has been verified in a number of special cases, eg., when the host graph is countable (see [1] for a survey of what is known), the general question is still unresolved.


Bibliography
1 R. Aharoni, A few remarks on a conjecture of Erdös on the infinite version of Menger's theorem, The Mathematics of Paul Erdös, II (R. L. Graham and J. Nešetril, eds.), 394-408, Springer-Verlag, Berlin, 1996.

2 D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.