Odd circuits in dense binary matroids

We show that, for each real number $α> 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge α2^{r(M)}$ 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.

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