A Sylvester-Gallai-type theorem for complex-representable matroids
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917651869073408 |
|---|---|
| author | Geelen, Jim Kroeker, Matthew E. |
| author_facet | Geelen, Jim Kroeker, Matthew E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03307 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Sylvester-Gallai-type theorem for complex-representable matroids Geelen, Jim Kroeker, Matthew E. Combinatorics 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. |
| title | A Sylvester-Gallai-type theorem for complex-representable matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2212.03307 |