Every simple rank-r matroid with no (t+1)-element independent flat has at least as many elements as the direct sum of t binary projective geometries of nearly equal ranks, with equality only for this matroid when r is at least 2t.
Tur´ an, On an extremal problem in graph theory, Matematikai ´ es Fizikai Lapok(1941) (in Hungarian), 436—452
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The smallest matroids with no large independent flat
Every simple rank-r matroid with no (t+1)-element independent flat has at least as many elements as the direct sum of t binary projective geometries of nearly equal ranks, with equality only for this matroid when r is at least 2t.