Intersections of circuits and cocircuits in binary matroids

Oxley has shown that if, for some k >_-4, a matroid M has a k-element set that is the intersection of a circuit and a cocircuit, then M has a 4-element set that is the intersection of a circuit and a cocircuit. We prove that, under the above hypothesis, for k I> 6, a binary matroid will also have a 6-element set that is the intersection of a circuit and a cocircuit. In addition, we determine explicitly the regular matroids which do not have a 6-element set that is the intersection of a circuit and cocircuit. Finally, we prove that in the case of graphs, if for some k 1> 4, a circuit and a cocircuit intersect in k elements, then there must be a circuit and a cocircuit that intersect in (k 2) elements. @ 1999 Elsevier Science B.V. All rights reserved