Cooperative coloring of matroids

Let $M_1,M_2,\dots ,M_k$ be a collection of matroids on the same ground set $E$. A coloring $c:E\to \{1,2,\dots ,k\}$ is called \emph{cooperative} if for every color $j$, the set of elements in color $j$ is independent in $M_j$. We prove that such coloring always exists provided that every matroid $M_j$ is itself $k$-colorable (the set $E$ can be split into at most $k$ independent sets of $M_j$). We derive this fact from a generalization of Seymour’s list coloring theorem for matroids, which asserts that every $k$-colorable matroid is $k$-list colorable, too. We also point on some consequences for the game-theoretic variants of cooperative coloring of matroids.