On the relative importance of excluded minors

If E is a set of matroids, then EX(E) denotes the set of matroids that have no minor isomorphic to a member of E. If E′⊆E, we say that E′ is superfluous if EX(E−E′)−EX(E) contains only finitely many 3-connected matroids. We determine the superfluous subsets of six well-known collections of excluded minors.