Odd circuits in dense binary matroids

TL;DR

This paper proves that dense binary matroids without small odd circuits have bounded critical number, using a regularity lemma for finite abelian groups, extending understanding of matroid structure.

Contribution

It establishes a bound on the critical number of dense binary matroids avoiding small odd circuits, applying Green's regularity lemma in a novel way.

Findings

Dense binary matroids with no small odd circuits have bounded critical number.

The proof uses a regularity lemma for finite abelian groups.

The result generalizes previous structural theorems in matroid theory.

Abstract

We show that, for each real number $\alpha > 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge \alpha 2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical number at most $c$. The result is an easy application of a regularity lemma for finite abelian groups due to Green.