A geometric version of the Andrásfai–Erdős–Sós Theorem

For each odd integer k≥5k≥5, we prove that, if M is a simple rank-r binary matroid with no odd circuit of length less than k   and with |M|>k2r−k+1|M|>k2r−k+1, then M is isomorphic to a restriction of the rank-r   binary affine geometry; this bound is tight for all r≥k−1r≥k−1. We use this to give a simpler proof of the following result of Govaerts and Storme: for each integer n≥2n≥2, if M is a simple rank-r   binary matroid with no PG(n−1,2)PG(n−1,2)-restriction and with |M|>(1−112n+2)2r, then M   has critical number at most n−1n−1. That result is a geometric analogue of a theorem of Andrásfai, Erdős and Sós in extremal graph theory.