Unavoidable Minors of Matroids with Minimum Cocircuit Size Four

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has K₅ or K2,2,2 as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which every cocircuit has size at least four has F₇, M^*(K3,3), M(K₅), or M(K2,2,2) as a minor. Generalizing these results, this paper proves that every simple matroid in which all cocircuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements.