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 ℓ, we prove that, if a matroid is colorable from a particular lists of size ℓ, then it is colorable from any lists of size ℓ. This gives an explicit necessary and sufficient condition for a matroid to be colorable from any lists of a fixed size.As an application, we show how to use our condition to derive several base exchange properties.