  We show that, for each real number $\epsilon > 0$ there is an integer $c$
such that, if $M$ is a simple triangle-free binary matroid with $|M| \ge
(\tfrac{1}{4} + \epsilon) 2^{r(M)}$, then $M$ has critical number at most $c$.
We also give a construction showing that no such result holds for any real
number less than $\tfrac{1}{4}$. This shows that the "critical threshold" for
the triangle is $\tfrac 1 4$. We extend the notion of critical threshold to
every simple binary matroid $N$ and conjecture that, if $N$ has critical number
$c\ge 3$, then $N$ has critical threshold $1-i\cdot 2^{-c}$ for some $i\in
\{2,3,4\}$. We give some support for the conjecture by establishing lower
bounds.
