On the number of matroids compared to the number of sparse paving matroids

Abstract

It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that limn!1 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 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 (formula presented) 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 all (formula presented) asymptotically almost all matroids on n elements have rank in the range (formula presented).