The smallest $I_5$-free and triangle-free binary matroids

TL;DR

This paper identifies the smallest simple binary matroids that are triangle-free and lack a five-element independent flat, addressing a specific case of a conjecture by Nelson and Norin.

Contribution

It determines the minimal size of certain binary matroids with specific properties, solving a particular case of an open conjecture.

Findings

Identified the smallest such binary matroids

Confirmed the conjecture for this special case

Provided structural insights into these matroids

Abstract

We determine the smallest simple triangle-free binary matroids that have no five-element independent flat. This solves a special case of a conjecture of Nelson and Norin.