Joint with Nathan Bowler and Daniel Slilaty.
We obtain asymptotically precise upper and lower bounds on the number of matroids of rank at least 4; a gap remains for rank-3 matroids.
Of particular interest are sparse paving matroids. These well-behaved matroids are related to both Steiner systems and matchings in hypergraphs. Existing enumeration methods (Linial-Luria, Bennett-Bohman) that work in those settings can be made to work for sparse paving matroids as well.
A combinatorial argument then extends these bounds to the broader class of paving matroids, and thence to general matroids.