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For each integer $n \geq 2$, we prove that, if $M$ is a simple rank-$r$$PG(n-1,2)$-free binary matroid with $|M|>\left(1-\frac{3}{2^n}\right)2^r$,then there is a triangle-free corank-$(n-2)$ flat of $M$.

Dense PG(n-1,2)-free binary matroids