The Erdös–Pósa property for matroid circuits

The number of disjoint cocircuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint cocircuits; consider, for example, M(Kn) and Un,2n. Also the bicircular matroids B(Kn) have arbitrarily large rank and have no 3 disjoint cocircuits. We prove that for each k and n there exists a constant c such that, if M is a matroid with rank at least c, then either M has k disjoint cocircuits or M contains a Un,2n-, M(Kn)-, or B(Kn)-minor.