On Minors Avoiding Elements in Matroids

Let ℱ be a collection of 3-connected matroids, none a proper minor of another, such that if M is a 3-connected matroid having a proper ℱ-minor and e is an element of M, then M has an ℱ-minor avoiding e. This paper proves that there are precisely two collections ℱ with this property: {U2,4} and {U2,4, M(K4)}. Several extensions of this result and some similar results for 2-connected matroids are also established