Almost all matroids are non-representable

TL;DR

This paper proves that as the size of matroids increases indefinitely, the likelihood of a matroid being representable approaches zero, highlighting the rarity of representability in large matroids.

Contribution

It establishes that almost all large matroids are non-representable, providing a probabilistic asymptotic result in matroid theory.

Findings

Proportion of representable matroids tends to zero as n increases

Most large matroids are non-representable

Asymptotic behavior of matroid representability

Abstract

We prove that, as $n$ approaches infinity, the proportion of $n$-element matroids that are representable tends to zero.