On the number of matroids

We consider the problem of determining m n , the number of matroids on n elements. The best known lower bound on m n is due to Knuth (1974) who showed that loglogm n is at least n - 3/2logn - O(1). On the other hand, Piff (1973) showed that loglogm n = n - logn + loglogn + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth’s bound. We show that this is indeed the case, and prove an upper bound on loglogm n that is within an additive 1+o(1) term of Knuth’s lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of stable sets in the Johnson graph to give a compressed representation of matroids.