List coloring of matroids and base exchange properties

A coloring of a matroid is an assignment of colors to the elements of its ground set. We restrict to proper colorings - those for which elements of the same color form an independent set. Seymour proved that a $k$-colorable matroid is also colorable from any lists of size $k$. We generalize this theorem to the case when lists have still fixed sizes, but not necessarily equal. For any fixed size of lists assignment $\mathrm{\ell }$, we prove that, if a matroid is colorable from a particular lists of size $\mathrm{\ell }$, then it is colorable from any lists of size $\mathrm{\ell }$. This gives an explicit necessary and sufficient condition for a matroid to be list colorable from any lists of a fixed size. As an application, we show how to use our condition to derive several base exchange properties.