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 M⧹e 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).