A Sylvester-Gallai-type theorem for complex-representable matroids

TL;DR

This paper extends the Sylvester-Gallai theorem to complex-representable matroids, showing that high-rank such matroids contain specific flat structures, with improved bounds and a simpler proof compared to previous work.

Contribution

It provides a new, more elementary proof for the existence of specific flats in complex-representable matroids, improving bounds over prior results.

Findings

High-rank complex-representable matroids contain rank-k flats with exactly k points.

The bounds for the rank of matroids guaranteeing such flats are improved.

The proof is more elementary than previous approaches.

Abstract

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.