Permutations minimizing the number of collinear triples
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929659589951488 |
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| author | Cooper, Joshua Hyatt, Jack |
| author_facet | Cooper, Joshua Hyatt, Jack |
| contents | We characterize the permutations of $\mathbb{F}_q$ whose graph minimizes the number of collinear triples and describe the lexicographically-least one, affirming a conjecture of Cooper-Solymosi. This question is closely connected to Dudeney's No-3-in-a-Line problem, the Heilbronn triangle problem, and the structure of finite plane Kakeya sets. We discuss a connection with complete sets of mutually orthogonal latin squares and state a few open problems primarily about general finite affine planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutations minimizing the number of collinear triples Cooper, Joshua Hyatt, Jack Combinatorics 51E15 (Primary) 05B25, 11T99 (Secondary) We characterize the permutations of $\mathbb{F}_q$ whose graph minimizes the number of collinear triples and describe the lexicographically-least one, affirming a conjecture of Cooper-Solymosi. This question is closely connected to Dudeney's No-3-in-a-Line problem, the Heilbronn triangle problem, and the structure of finite plane Kakeya sets. We discuss a connection with complete sets of mutually orthogonal latin squares and state a few open problems primarily about general finite affine planes. |
| title | Permutations minimizing the number of collinear triples |
| topic | Combinatorics 51E15 (Primary) 05B25, 11T99 (Secondary) |
| url | https://arxiv.org/abs/2501.02331 |