A Sharp Bound on the Size of a Connected Matroid

This paper proves that a connected matroid M in which a largest circuit and a largest cocircuit have c and c∗ elements, respectively, has at most 1 2 cc∗ elements. It is also shown that if e is an element of M and ce and ce are the sizes of a largest circuit containing e and a largest cocircuit containing e, then |E(M)| ≤ (ce−1)(ce−1)+1. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman’s width-length inequality which asserts that the former inequality can be reversed for regular matroids when ce and ce are replaced by the sizes of a smallest circuit containing e and a smallest cocircuit containing e. Moreover, it follows from the second inequality that if u and v are distinct vertices in a 2-connected loopless graph G, then |E(G)| cannot exceed the product of the length of a longest (u, v)-path and the size of a largest minimal edge-cut separating u from v.