Almost all matroids are paving

Establish that the asymptotic proportion of matroids on n-element ground sets that are paving matroids tends to 1 as n goes to infinity, i.e., prove that the fraction of matroids that are paving converges to 1.

Background

A paving matroid of rank r is a matroid whose circuits all have size r or r+1. In the paper, paving matroids are highlighted because they exhibit no monotonicity violations in the matroid bingo framework: circuits of maximum size have strictly smaller winning probability than any smaller circuit, and paving matroids have only those two circuit sizes.

The authors recall a well-known conjecture in matroid theory asserting that, asymptotically, almost all matroids are paving. This long-standing conjecture underpins their subsequent conjecture about monotonicity holding for almost all matroids.