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- W2952739968 abstract "It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that $lim_{nrightarrowinfty} s_n/m_n = 1$, 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 $$lim_{nrightarrow infty}frac{log s_n}{log m_n}=1.$$ 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 $n/2 pm betasqrt{n}$." @default.
- W2952739968 created "2019-06-27" @default.
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- W2952739968 date "2014-11-04" @default.
- W2952739968 modified "2023-09-27" @default.
- W2952739968 title "On the number of matroids compared to the number of sparse paving matroids" @default.
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