Maximum size binary matroids with no AG(3,2)-minor are graphic

Abstract

We prove that the maximum size of a simple binary matroid of rank $r\ge 5$ with no AG(3,2)-minor is $\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)$ and characterise those matroids achieving this bound. When $r\ge 6$, 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.