Abstract
A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroidM has unavoidable structure. For every n > 2, there is an integer f (n) so that if |E(M)| > f (n), thenM has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K3,n, or U2,n or Un−2,n. In this paper, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)| > g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K1,1,1,n, a specific single-element extension ofM(K3,n) or the dual of this extension, or U2,n or Un−2,n