  We study simplicial complexes (hypergraphs closed under taking subsets) that
are the intersection of a given number k of matroids. We prove bounds on their
chromatic numbers (the minimum number of edges required to cover the ground
set) and their list chromatic numbers. Settling a conjecture of Kiraly and
Berczi et. al., we prove that the list chromatic number is at most k times the
chromatic number. Following the footsteps of Edmonds, who considered the case
k=2, we study three polytopes associated with k-tuples of matroids, and prove
bounds on the distances between them. The tools used are in part topological.
