A generalization of a graph result of D.W. Hall Original Research Article

D.W. Hall proved that every simple 3-connected graph with a K5-minor must have a K3.3-minor, the only exception being K5 itself. In this paper, we prove that every 3-connected binary matroid with an M(K5)-minor must have an M(K3.3)- or M*(K3.3)-minor, the only exceptions being M(K5), a highly symmetric 12-element matroid which we call T12, and any single-element contraction of T12.