intersection (where the intersection ranges over all finite sets ) is itself an infinite set , then every finite simple path that ends in a vertex of can be extended to reach an additional vertex of , and repeating this extension process produces a ray passing through infinitely many vertices of . This ray determines the given haven. On the other hand, if is finite, then (by working in the subgraph )it can be assumed to be empty. In this case, for each finite set of vertices there is a finite set with the property that is disjoint from . If a robber follows the evasion strategy determined by the haven, and the police follow a strategy given by this sequence of sets, then the path followed by the robber forms a ray that determines the haven. Thus, every equivalence class of rays defines a unique haven, and every haven is defined by an equivalence class of rays. For any cardinal number , an infinite graph has a haven of order if and only if it has a clique minor of order . That is, for uncountable cardinalities, the largest order of a haven in is the Hadwiger number of . References Graph theory objects Graph minor theory Pursuit–evasion