  The Sylvester-Gallai Theorem states that every rank-$3$ real-representable
matroid has a two-point line. We prove that, for each $k\ge 2$, every
complex-representable matroid with rank at least $4^{k-1}$ has a rank-$k$ flat
with exactly $k$ points. For $k=2$, this is a well-known result due to Kelly,
which we use in our proof. A similar result was proved earlier by Barak, Dvir,
Wigderson, and Yehudayoff and later refined by Dvir, Saraf, and Wigderson, but
we get slightly better bounds with a more elementary proof.
