Maximum Size Binary Matroids with no $AG(3,2)$-Minor are Graphic
Joseph P. S. Kung, Dillon Mayhew, Irene Pivotto, Gordon Royle · SIAM Journal on Discrete Mathematics · 2014
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 characterize 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 matroids of lower ranks meeting or exceeding this bound. In addition, we determine the size function for nongraphic simple binary matroids with no $AG(3,2)$-minor and characterize the matroids of maximum size for each rank.