On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)?5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)?B, then |E(M)|3r(M)?4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)?c(M).