Counting matroids in minor-closed classes

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,kU2,k, U3,6U3,6, P6P6, Q6Q6 or R6R6, 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 an M(K4)M(K4)-minor which asymptotically matches the best known lower bound on the number of all matroids, due to Knuth.