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 1-1 to sparse paving matroids on $n$ elements.

Original language | English |
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Publisher | s.n. |
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Number of pages | 12 |
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Publication status | Published - 2014 |
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Name | arXiv.org |
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Volume | 1411.0935 [math.CO] |
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