Excluding Infinite Clique Minors

Two of the authors proved a well-known conjecture of K. Wagner, that in any infinite set of finite graphs there are two graphs so that one is a minor of the other. A key lemma was a theorem about the structure of finite graphs that have no $K_n$ minor for a fixed integer $n$. Here, the authors obtain an infinite analog of this lemma---a structural condition on a graph, necessary and sufficient for it not to contain a $K_n$ minor, for any fixed infinite cardinal $n$.