Coloring matroids (Q1356738)

``Good'' colorings of a matroid partition the ground set \(E=E_1\cup E_2\cup\cdots\cup E_c\) such that every transversal (\(T\subseteq E\) with \(|T\cap E_i|\leq 1\)) is independent, and such that for every line \(\ell\subseteq E\) some \(\ell\cap E_i\) is a singleton. This notion is due to Falk and Jambu [unpublished preprint, 1989]. Here it is proved that every good coloring of \(M\) uses exactly \(c=r(M)\) colors. The same was proved by \textit{H. Terao} [Adv. Math. 91, No. 1, 45-53 (1992; Zbl 0772.05029)] for the stronger concept of ``nice'' colorings, where for every non-empty flat \(F\subseteq E\) some \(F\cap E_i\) is a singleton. Nice colorings are of interest since they characterize certain types of factorizations of Orlik-Solomon algebras.