Odd hypergraph Mantel theorems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908595173457920 |
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| author | Hou, Jianfeng Liu, Xizhi Zhang, Yixiao Zhao, Hongbin Zhu, Tianming |
| author_facet | Hou, Jianfeng Liu, Xizhi Zhang, Yixiao Zhao, Hongbin Zhu, Tianming |
| contents | A classical result of Sidorenko (1989) shows that the Turán density of every $r$-uniform hypergraph with three edges is bounded from above by $1/2$. For even $r$, this bound is tight, as demonstrated by Mantel's theorem on triangles and Frankl's theorem on expanded triangles. In this note, we prove that for odd $r$, the bound $1/2$ is never attained, thereby answering a question of Keevash and revealing a fundamental difference between hypergraphs of odd and even uniformity. Moreover, our result implies that the expanded triangles form the unique class of three-edge hypergraphs whose Turán density attains $1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Odd hypergraph Mantel theorems Hou, Jianfeng Liu, Xizhi Zhang, Yixiao Zhao, Hongbin Zhu, Tianming Combinatorics A classical result of Sidorenko (1989) shows that the Turán density of every $r$-uniform hypergraph with three edges is bounded from above by $1/2$. For even $r$, this bound is tight, as demonstrated by Mantel's theorem on triangles and Frankl's theorem on expanded triangles. In this note, we prove that for odd $r$, the bound $1/2$ is never attained, thereby answering a question of Keevash and revealing a fundamental difference between hypergraphs of odd and even uniformity. Moreover, our result implies that the expanded triangles form the unique class of three-edge hypergraphs whose Turán density attains $1/2$. |
| title | Odd hypergraph Mantel theorems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.10590 |