Counting matroids in minor-closed classes

TL;DR

This paper introduces the concept of cover complexity to measure matroid complexity and uses it to analyze the asymptotic size of minor-closed classes of matroids, confirming some cases of a conjecture and providing bounds.

Contribution

It defines cover complexity for matroids and applies it to prove asymptotic smallness of certain minor-closed classes, advancing understanding of matroid enumeration.

Findings

Matroids without specific minors are asymptotically small in number.

Cover complexity bounds help classify matroid classes.

Lower bounds match known bounds for all matroids.

Abstract

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without $M(K_4)$-minor which asymptoticaly matches the best known lower bound on the number of all matroids, due to Knuth.