A Sylvester–Gallai-Type Theorem for Complex-Representable Matroids

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.