The countable Erdős-Menger conjecture with ends (Q1403915)

Erdős conjectured that, given an infinite graph \(G\) and vertex sets \(A, B\subseteq V(G)\), there exist a set \(\mathcal P\) of disjoint \(A\)-\(B\) separators \(X\) `on' \(\mathcal P\), in the sense that \(X\) consists of a choice of one vertex from each path in \(\mathcal P\). This paper proves, for countable graphs \(G\), the extension of this conjecture in which \(A, B\) and \(X\) are allowed to contain edges as well as vertices, and where the closure of \(A\) avoids \(B\) and vice versa. Note that without the closure condition the extended conjecture is false.