Doubly exponentially many Ingleton matroids

TL;DR

This paper proves that the number of Ingleton matroids on a ground set of size n grows doubly exponentially with n, indicating that most Ingleton matroids are not representable.

Contribution

It establishes the asymptotic count of Ingleton matroids, showing their abundance and the predominance of non-representable cases.

Findings

Number of Ingleton matroids on [n] is doubly exponential in n.

Almost all Ingleton matroids are non-representable.

Ingleton's inequality characterizes a large class of matroids.

Abstract

A matroid is Ingleton if all quadruples of subsets of its ground set satisfy Ingleton's inequality. In particular, representable matroids are Ingleton. We show that the number of Ingleton matroids on ground set $[n]$ is doubly exponential in $n$; it follows that almost all Ingleton matroids are non-representable.