Branch-Width and Rota's Conjecture

We begin by giving some background to this result. A matroid M is an excluded minor for a minor-closed class of matroids if M is not in the class but all proper minors of M are. It is natural to attempt to characterize a minor-closed class of matroids by giving a complete list of its excluded minors. Unfortunately, a minor-closed class of matroids can have an infinite number of excluded minors. For example, this is the case for the matroids representable over the rationals; see [7]. This contrasts strikingly with the fundamental theorem of Robertson and Seymour which says that graphs are