Inclusion of forbidden minors in random representable matroids

Our results directly imply that M[An] is a.a.s. not contained in any proper, minor-closed class M of Fq-representable matroids, provided: (i) n−m(n)→∞, and (ii) m(n) is at least the minimum rank of any Fq-representable forbidden minor of M, for all sufficiently large n. As an application, this shows that graphic matroids are a vanishing subset of linear matroids, in a sense made precise in the paper. Our results provide an approach for applying the rich theory around matroid minors to the less-studied field of random matroids.