  We prove that the maximum size of a simple binary matroid of rank $r \geq 5$
with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids
achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the
unique matroid meeting the bound, but there are a handful of smaller examples.
In addition, we determine the size function for non-regular simple binary
matroids with no AG(3,2)-minor and characterise the matroids of maximum size
for each rank.
