On the structure of dense triangle-free binary matroids

TL;DR

This paper provides a detailed structural characterization of dense triangle-free binary matroids, showing that sufficiently large such matroids have a critical number of at most 2.

Contribution

It offers an exact structural description of dense triangle-free binary matroids and establishes a size threshold for their critical number.

Findings

Dense triangle-free binary matroids with size above a specific threshold have critical number at most 2.

The paper characterizes the structure of such matroids precisely.

It advances understanding of the interplay between density, triangle-freeness, and critical number in binary matroids.

Abstract

We prove, by means of an exact structural description, that every simple triangle-free binary matroid $M$ with $|M| > \tfrac{33}{128}2^{r(M)}$ has critical number at most $2$.