Triples in Matroid Circuits

Watkins and Mesner characterized edge-triples of a graph which are not in any circuit, and Chakravarti and Robertson solved the dual problem of edge-triples not in a bond. Here we give a common generalization, solving the problem for all binary matroids. Our main result is that if e, f, g are elements of a 3-connected, internally 4-connected binary matroid, then there is a circuit containing e, f, g, unless either {e, f, g} is a cocircuit, or the matroid is graphic and e, f, g are edges of the graph with a common vertex. The more general non-binary problem is still open.