For a matroid M having m rank-one flats, the density d(M) is
r(M)m unless m=0, in which case d(M)=0. A matroid is
density-critical if all of its proper minors of non-zero rank have lower
density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among
simple matroids that cannot be covered by k independent sets is
density-critical. It is straightforward to show that U1,k+1 is the only
minor-minimal loopless matroid with no covering by k independent sets. We
prove that there are exactly ten minor-minimal simple obstructions to a matroid
being able to be covered by two independent sets. These ten matroids are
precisely the density-critical matroids M such that d(M)>2 but d(N)≤2 for all proper minors N of M. All density-critical matroids of density
less than 2 are series-parallel networks. For k≥2, although finding all
density-critical matroids of density at most k does not seem straightforward,
we do solve this problem for k=49.Comment: 16 page