Turán density of tight cycles minus one edge in the $\ell_2$-norm
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913921438318592 |
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| author | Bodnár, Levente Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhao, Hongbin |
| author_facet | Bodnár, Levente Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhao, Hongbin |
| contents | The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv 0\pmod3$, every $C_{\ell}^{3-}$-free $3$-graph whose $\ell_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh--Clemen--Lidický~[Surveys in combinatorics 2022, 21-63] in a stronger form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turán density of tight cycles minus one edge in the $\ell_2$-norm Bodnár, Levente Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhao, Hongbin Combinatorics The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv 0\pmod3$, every $C_{\ell}^{3-}$-free $3$-graph whose $\ell_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh--Clemen--Lidický~[Surveys in combinatorics 2022, 21-63] in a stronger form. |
| title | Turán density of tight cycles minus one edge in the $\ell_2$-norm |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.00812 |