Intersections of circuits and cocircuits in binary matroids (Q1296980)

Oxley has shown that if, for some \(k\geq 4\), a matroid \(M\) has a \(k\)-element set that is the intersection of a circuit and cocircuit, then \(M\) has a 4-element set that is the intersection of a circuit and a cocircuit. In the paper, it is proved that, under the above hypothesis, for \(k\geq 6\), a binary matroid also has a 6-element set that is the intersection of a circuit and a cocircuit. In addition, the regular matroids which do not have a 6-element set that is the intersection of a circuit and cocircuit are determined. Finally, in the case of graphs, it is shown that, if, for some \(k\geq 4\), a circuit and a cocircuit intersect in \(k\) elements, then there must be a circuit and a cocircuit that intersect in \((k-2)\) elements.