Paving matroid

A matroid M is called paving if every set of size r(M)-1 is independent. It is called sparse paving if in addition every circuit of size r(M) is closed. The importance of (sparse) paving matroids stems from their large number: lower bounds on the number of matroids are obtained by counting sparse paving matroids, see e.g. Knuth . In fact, Mayhew et al. conjecture that if $$m_n$$ is the number of matroids and $$s_n$$ is the number of sparse paving matroids on $$n$$ elements, then $$\lim_{n \to \infty} \frac{m_n}{s_n} =1$$ . Progress towards this conjecture has been made in .

See also Wikipedia:Paving matroid.