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 ≥ 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...

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Bibliographic Details
Published in:Discrete & computational geometry 2025-01, Vol.73 (1), p.258-263
Main Authors: Geelen, Jim, Kroeker, Matthew E.
Format: Article
Language:English
Subjects:
Combinatorics
Computational Mathematics and Numerical Analysis
Geometry
Linear algebra
Mathematics
Mathematics and Statistics
Terminology
Theorems
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Summary:The Sylvester–Gallai Theorem states that every rank-3 real-representable matroid has a two-point line. We prove that, for each k ≥ 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.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-024-00661-x