Asymptotic prevalence of paving matroids

Determine whether the limit of the proportion of matroids on n elements that are paving matroids (equivalently sparse paving matroids) equals 1 as n tends to infinity. This problem seeks to quantify the asymptotic abundance of paving matroids within the class of all matroids and has implications for coding-theoretic applications since many results in the paper hinge on properties of paving and sparse paving matroids.

Background

Paving matroids are those in which every circuit has size equal to the rank or rank plus one, and sparse paving matroids are paving matroids whose duals are also paving. These classes play a central role in the paper because the generalized Hamming weights of paving matroids form subadditive sequences, and for sparse paving matroids these sequences are extended subadditive, leading to sharp formulas for symbolic powers and Waldschmidt constants.

Understanding how common paving matroids are among all matroids is important both combinatorially and for applications in coding theory. The cited conjecture posits that, asymptotically, almost all matroids are paving, which would suggest that the favorable properties established in the paper for paving and sparse paving matroids may be generically applicable.