Coloring the intersection of two matroids

A result {matcomp} from 2006 of Aharoni and the first author of this paper states that for any two natural numbers p, q, where p divides q, if a matroid M is p-colorable and a matroid N is q-colorable then M ∩ N is (p+q)-colorable. In this paper we show that the assumption that p divides q is in fact redundant, and we also prove that M ∩ N is even p+q list-colorable. The result uses topology and relies on a new parameter yielding a lower bound for the topological connectivity of the intersection of two matroids.