On the Number of Bases of Almost All Matroids

Abstract

For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that

$$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$

for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U_k,2k-minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity $\geq\Omega\;{(\sqrt {log(n)})}$, and (4) do not arise as the truncation of another matroid.

Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.