Extensions and Deletions of matroid classes closed under flats

Abstract

We call a class of matroids hereditary if it is closed under restriction to flats. For a hereditary class $\mathcal{M}$, its extension class consists of all matroids in $\mathcal{M}$ together with their single-element extensions. The deletion class consists of all matroids in $\mathcal{M}$ along with their single-element deletions. We prove that if $\mathcal{M}$ has finitely many forbidden flats, then the forbidden flats for its extension class have bounded rank. For $GF(q)$-representable matroids where $q$ is in $\{2,3\}$, we exploit correspondence with $2$-colorings of projective geometries to establish the analogous result for the deletion class. We also note the consequences for hereditary classes of graphs, discussing the interplay of graphs and matroids.