Capturing matroid elements in unavoidable 3-connected minors

A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n > 2 , there is an integer f ( n ) so that if | E ( M ) | > f ( n ) , then M has a minor isomorphic to the rank- n wheel or whirl, a rank- n spike, the cycle or bond matroid of K 3 , n , or U 2 , n or U n − 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 K 1 , 1 , 1 , n , a specific single-element extension of M ( K 3 , n ) or the dual of this extension, or U 2 , n or U n − 2 , n .