RDÖS - P ÓSA PROPERTY FOR MATROID CIRCUITS by Jim Geelen and Kasper

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