Characterising k-connected sets in infinite graphs

A k-connected set in an infinite graph, where k > 0 is an integer, is a set of vertices such that any two of its subsets of the same size ≤ k can be connected by disjoint paths in
the whole graph. We characterise the existence of k-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such k-connected sets: if a graph contains no such k-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a k-connected set.