It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that limn→∞ sn/mn = 1, where mn denotes the number of matroids on n elements, and sn the number of sparse paving matroids. In this paper, we show that lim n→∞ log sn log mn = 1.We prove this by arguing that each matroid on n elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on n elements. As a consequence of our result, we find that for some β > 0, asymptotically almost all matroids on n elements have rank in the range n/2 ± β √ n.2

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that s(n) ∼ m(n), where m(n) denotes the number of matroids on a fixed groundset of size n, and s(n) the number of sparse paving matroids. In an earlier paper, we showed that log s(n) ∼ log m(n). The bounds that we used for that result were dominated by matroids of rank r ≈ n/2. In this paper we consider the relation between the number of sparse paving matroids s(n, r) and the number of matroids m(n, r) on a fixed groundset of size n of fixed rank r. In particular, we show that log s(n, r) ∼ log m(n, r) whenever r ≥ 3, by giving asymptotically matching upper and lower bounds.Our upper bound on m(n, r) relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid.We also obtain that the collection of essential flats and their ranks gives a concise description of matroids.

For a matroid M of rank r on n elements, let b(M ) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show thatfor asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U k,2k -minor, whenever k ≤ O(log(n)), (2) have girth ≥ Ω(log(n)), (3) have Tutte connectivity ≥ Ω( log(n)), and (4) do not arise as the truncation of another matroid.Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.Conjecture 1.1. As n → ∞, asymptotically almost all matroids on n elements are sparse paving.If this conjecture were true, then several other asymptotic properties of matroids would follow with little extra work, as it would suffice to establish the property for almost all sparse paving matroids. For example, almost all sparse paving matroids contain a fixed uniform matroid, are highly connected, have high girth, and the analogous statements for general matroids would follow. We next describe our intuition supporting Conjecture 1.1, which we partially substantiated to obtain the results of this paper.Consider the Johnson graph J(E, r), whose vertices are the subsets of E of cardinality r, and in which two vertices X, Y are adjacent exactly if |X ∩ Y | = r − 1. The Johnson graph will serve as an 'ambient space' for all the matroids on ground set E and of rank r. In what follows, we will write G := J(E, r).

We prove that asymptotically almost all matroids have a trivial automorphism group, or an automorphism group generated by a single transposition. Additionally, we show that asymptotically almost all sparse paving matroids have a trivial automorphism group.