  It has been conjectured that sparse paving matroids will eventually
predominate in any asymptotic enumeration of matroids, i.e. that
$\lim_{n\rightarrow\infty} s_n/m_n = 1$, where $m_n$ denotes the number of
matroids on $n$ elements, and $s_n$ the number of sparse paving matroids. In
this paper, we show that $$\lim_{n\rightarrow \infty}\frac{\log s_n}{\log
m_n}=1.$$ 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 $\beta > 0$,
asymptotically almost all matroids on $n$ elements have rank in the range $n/2
\pm \beta\sqrt{n}$.
