The coloring game on matroids

We prove an upper bound χg(M)≤2χ(M) for every matroid M. This improves and extends a result of Bartnicki, Grytczuk and Kierstead [1], who showed that χg(M)≤3χ(M) holds for graphic matroids. Our bound is almost tight, as we construct a family of matroids Mk (for k≥3) satisfying χ(Mk)=k and χg(Mk)=2k−1, which improves a construction of Bartnicki et al. by 1.