Paving matroids that are not sparse paving

TL;DR

This paper investigates the difference between paving and sparse paving matroids, showing that non-sparse paving matroids are significantly numerous, and introduces constructions based on hyperplanes and stable sets.

Contribution

It establishes a lower bound on the number of paving matroids that are not sparse paving, revealing their substantial size and providing new construction methods.

Findings

Non-sparse paving matroids are logarithmically large.

A lower bound relating paving and sparse paving matroids is proven.

Construction methods involve hyperplanes and stable sets in Johnson graphs.

Abstract

The Mayhew--Newman--Welsh--Whittle conjecture predicts that asymptotically almost all matroids are sparse paving. We study the gap between paving and sparse paving matroids at the logarithmic scale. Let \(p_n\) be the number of paving matroids on \([n]\), let \(sp_n\) be the number of sparse paving matroids on \([n]\), and let \(sp_{n,r}\) be the number of rank-\(r\) sparse paving matroids on \([n]\). We prove that \[ p_n-sp_n\ge sp_{n,\lfloor n/2\rfloor}^{1-o(1)}. \] Thus the paving matroids that are not sparse paving are themselves logarithmically large. The construction prescribes one hyperplane larger than the rank and then counts stable sets in an induced subgraph of a Johnson graph. We also give amplified versions obtained by varying the large hyperplane and by prescribing distance-six families of large hyperplanes.