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

TL;DR

This paper proves that the logarithm of the number of sparse paving matroids asymptotically matches that of all matroids, supporting the conjecture that sparse paving matroids dominate as the number of elements grows.

Contribution

It establishes that the logarithmic ratio of sparse paving matroids to all matroids approaches 1, providing a key step towards the conjecture that sparse paving matroids predominate.

Findings

 log s_n / \u00a9 log m_n approaches 1 as n increases

Most matroids on n elements have rank close to n/2

Each matroid can be described by a stable set in a Johnson graph

Abstract

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 $n/2 \pm \beta\sqrt{n}$.