Disjoint cocircuits in matroids with large rank
To gain insight into minor-closed classes of GF\((q)\)-representable matroids that omit a given characteristic, the authors prove the following theorem: For any positive integers \(n, k\), and \(q\), there exists an integer \(R\) such that, if \(M\) is a matroid with no \(M(K_n)\)- or \(U_{2, q+2}\)-minor, then either \(M\) has a collection of \(k\) disjoint cocircuits or \(M\) has rank at most \(R\). Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdős-Pósa theorem [see \textit{P. Erdős} and \textit{L. Pósa}, Publ. Math. 9, 3-12 (1962; Zbl 0133.16701)].
scientific article; zbMATH DE number 1970691