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arXiv · 2510.10590

Odd hypergraph Mantel theorems

Jianfeng Hou·Xizhi Liu·Yixiao Zhang·Hongbin Zhao·Tianming Zhu
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Abstract

A classical result of Sidorenko (1989) shows that the Tur\'{a}n 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 Tur\'{a}n density attains $1/2$.

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Jianfeng Hou, Xizhi Liu, Yixiao Zhang, Hongbin Zhao, Tianming Zhu. 2025-10-12. Odd hypergraph Mantel theorems. https://arxiv.org/abs/2510.10590

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