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

© 2014 Society for Industrial and Applied Mathematics. We prove that the maximum size of a simple binary matroid of rank r ≥ 5 with no AG(3, 2)-minor is (2 r+1) and characterize those matroids achieving this bound. When r ≥ 6, the graphic matroid M (Kr+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.