Unavoidable Minors of Large 3-Connected Matroids

Abstract

This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,…,n−1}, the union of every set ofkof these lines has rankk+1.