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A Sylvester-Gallai-type theorem for complex-representable matroids. (arXiv:2212.03307v1 [math.CO]) http://arxiv.org/abs/2212.03307

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.

arxiv.org
December 8, 2022 at 3:10 AM · · feed2toot · 0 · 0 · 0
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