Unavoidable minors of large 3-connected matroids (Q1386436)

An earlier result of the same authors for 3-connected matroids is extended to the more general class of all 3-connected matroids. It is shown that for every integer \(n>2\) there exists a number \(N(n)\) such that every 3-connected matroid with at least \(N(n)\) elements has a minor isomorphic to an \((n+2)\)-point line or its dual, the cycle or cocycle matroid of \(K_{3,n}\), the cycle matroid of a wheel with \(n\) spokes, a whirl of rank \(n\), or a (uniform) \(n\)-spike. Introducing the idea of a hamiltonian partial representation of matroids which have a spanning circuit and using the fact that, up to duality, every sufficiently large 3-connected matroid has a big 3-connected hamiltonian minor, the proof of this theorem can be built on the earlier results for 3-connected binary matroids based on their matrix representations and some Ramsey-theoretic results for matrices. Furthermore it is shown that the above-mentioned result is best possible in the sense that the given list of matroids contains no redundancy.