Towards Unavoidable Minors of Binary 4-connected Matroids

We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.