Minors of a random binary matroid

TL;DR

This paper investigates the conditions under which a random binary matroid, generated from a matrix with columns of fixed weight, almost surely contains any fixed binary matroid as a minor when the matrix dimensions grow large.

Contribution

It establishes that for sufficiently large ratios of columns to rows and fixed column weight, the random binary matroid almost surely contains any fixed binary matroid as a minor.

Findings

For large m/n and k, the random matroid contains any fixed binary matroid as a minor.

As n approaches infinity, the probability approaches one that the matroid contains the fixed minor.

The result connects random matrix properties with matroid minor containment.

Abstract

Let ${\bf A}={\bf A}_{n,m,k}$ be a random $n\times m$ matrix over $\mathbf{GF}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m/n$ and $k$ are sufficiently large then as $n\to\infty$ the binary matroid induced by {\bf A} contains $M$ as a minor.