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

Rudi Pendavingh¹ ; Jorn van der Pol¹

¹Eindhoven University of Technology

The electronic journal of combinatorics, Tome 22 (2015) no. 2

Résumé

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 all $\beta > \displaystyle{\sqrt{\frac{\ln 2}{2}}} = 0.5887\cdots$, asymptotically almost all matroids on $n$ elements have rank in the range $n/2 \pm \beta\sqrt{n}$.

Détail

DOI : 10.37236/4899

Classification : 05B35, 05A16
Mots-clés : matroid theory, asymptotic enumeration