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

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 $n/2 \pm \beta\sqrt{n}$.Comment: 12 pages, 2 figure

Asymptotics of Symmetry in Matroids

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.Comment: 10 page

The matroid secretary problem for minor-closed classes and random matroids

We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.Comment: 15 pages, 0 figure

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

t has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that limn→∞sn/mn=1limn→∞sn/mn=1, where mnmn denotes the number of matroids on nn elements, and snsn the number of sparse paving matroids. In this paper, we show that limn→∞logsnlogmn=1. limn→∞log⁡snlog⁡mn=1. We prove this by arguing that each matroid on nn 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 nn elements. As a consequence of our result, we find that for all β>ln22−−−−√=0.5887⋯β>ln⁡22=0.5887⋯, asymptotically almost all matroids on nn elements have rank in the range n/2±βn−−√n/2±βn