On matroids without a non-Fano minor (Q1301675)

The family of matroids that do not contain the non-Fano matroid or its dual as a minor, are studied in the paper. It is proved that for any connected matroid \(M\), there are just finitely many minor-minimal matroids in the considered family that contain both an \(M\)-minor and a \(U_{2,4}\)-minor (see \textit{J. G. Oxley} [Matroid theory (1992; Zbl 0784.05002)] for notions and terminology). The paper is related to the intertwinning problem.