Typical Subgraphs of 3- and 4-Connected Graphs

We prove that, for every positive integer k, there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k-spoke wheel or K3,k; and that every internally 4-connected graph with at least N vertices has a minor isomorphic to the 2k-spoke double wheel, the k-rung circular ladder, the k-rung Mِbius ladder, or K4,k. We also prove an analogous result for infinite graphs.