Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n) so that if |E(M)|\u3ef(n), then M has a minor isomorphic to one of the rank-n wheel, the rank-n tipless binary spike, or the cycle or bond matroid of K3 n. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n) so that if |E(M)|\u3eg(n) and xεE(M), then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K11,1,n. In this paper, we prove that, for each i in {2,3}, there is an integer hi(n) so that if |E(M)|\u3ehi(n) and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z. © 2012 Elsevier Inc