Unavoidable Connected Matroids Retaining a Specified Minor

A sufficiently large connected matroid M contains a big circuit or a big cocircuit. Pou-Lin Wu showed that we can ensure that M has a big circuit or a big cocircuit containing any chosen element of M . In this paper, we begin with a fixed connected matroid N and we take M to be a connected matroid that has N as a minor. Our main result establishes that ifM is sufficiently large, then, up to duality, either M has a big connected minor in which N is a spanning restriction and the deletion of E(N) is a large connected uniform matroid, or M has as a minor the 2-sum of a big circuit and a connected single-element extension or coextension of N . In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.