  A coloring of a matroid is proper if elements of the same color form an
independent set. A theorem of Seymour asserts that a k-colorable matroid is
also colorable from any lists of size k. In this note we generalize this
theorem to the on-line setting. We prove that a coloring of a matroid from
lists of size k is possible even if appearances of colors in the lists are
recovered color by color by an adversary, while our job is to assign a color
immediately after it is recovered. We also prove a more general weighted
version of our result with lists of varying sizes. In consequence we get a
simple necessary and sufficient condition for matroid list colorability in
general case. The main tool we use is the multiple basis exchange property,
which we give a simple proof.
