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 U-2,U-k, U-3,U-6, P-6, Q(6) or R-6, 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(K-4)-minor which asymptotically matches the best known lower bound on the number of all matroids, due to Knuth. (C) 2014 Elsevier Inc. All rights reserved.