A Sylvester-Gallai-type theorem for complex-representable matroids (Q6662774)

The authors aim to generalize the Sylvester-Gallai Theorem which states: ``Given any finite set of points in the real plane, not all on a line, there is a line in the plane that contains exactly two of them''. Kelly extended these results to complex-representability via the theorem: Every rank-4 complex-representable matroid has a two-point line. The authors prove a generalization of Kelly's Theorem using more elementary methods than previous generalizations. The main theorem of the paper is: For each integer \(k \ge 2\), every complex-representable matroid with rank at least \(4(k-1)\) has an ordinary rank-\(k\) flat.