A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on n elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an N-minor is
asymptotically small in case N is one of the sparse paving matroids
U2,kβ, U3,6β, P6β, Q6β, or R6β, thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without M(K4β)-minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure