On the structure of extremal point-line arrangements
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915532925566976 |
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| author | Currier, Gabriel Solymosi, Jozsef Yu, Hung-Hsun Hans |
| author_facet | Currier, Gabriel Solymosi, Jozsef Yu, Hung-Hsun Hans |
| contents | In this note, we show that extremal Szemerédi-Trotter configurations are rigid in the following sense: If $P,L$ are sets of points and lines determining at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$ of points of size at most $k = k_0(C)$ such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth-Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester-Gallai configurations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_06115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of extremal point-line arrangements Currier, Gabriel Solymosi, Jozsef Yu, Hung-Hsun Hans Combinatorics 52C30 In this note, we show that extremal Szemerédi-Trotter configurations are rigid in the following sense: If $P,L$ are sets of points and lines determining at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$ of points of size at most $k = k_0(C)$ such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth-Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester-Gallai configurations. |
| title | On the structure of extremal point-line arrangements |
| topic | Combinatorics 52C30 |
| url | https://arxiv.org/abs/2409.06115 |