# Paving matroid

From Egres Open

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 ^{[1]}. In fact, Mayhew et al. ^{[2]} conjecture that if [math]m_n[/math] is the number of matroids and [math]s_n[/math] is the number of sparse paving matroids on [math]n[/math] elements, then [math]\lim_{n \to \infty} \frac{m_n}{s_n} =1[/math]. Progress towards this conjecture has been made in ^{[3]}.

See also Wikipedia:Paving matroid.

## References

- ↑ D.E. Knuth,
*The asymptotic number of geometries*, DOI link - ↑ D. Mayhew, M. Newman, D. Welsh, and G. Whittle,
*On the asymptotic proportion of connected matroids*, DOI link, author link - ↑ N. Bansal, R.A. Pendavingh, J.G. van der Pol,
*On the number of matroids*, arXiv link