{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T20:44:38Z","timestamp":1776977078799,"version":"3.51.4"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $\\beta &gt; \\displaystyle{\\sqrt{\\frac{\\ln 2}{2}}} = 0.5887\\cdots$, asymptotically almost all matroids on $n$ elements have rank in the range $n\/2 \\pm \\beta\\sqrt{n}$.<\/jats:p>","DOI":"10.37236\/4899","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:42:15Z","timestamp":1578699735000},"source":"Crossref","is-referenced-by-count":14,"title":["On the Number of Matroids Compared to the Number of Sparse Paving Matroids"],"prefix":"10.37236","volume":"22","author":[{"given":"Rudi","family":"Pendavingh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorn","family":"Van der Pol","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2015,6,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p51\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p51\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:15:39Z","timestamp":1579256139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i2p51"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,15]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/4899","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,15]]},"article-number":"P2.51"}}