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arXiv · 2507.00812

Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm

Levente Bodnár·Jinghua Deng·Jianfeng Hou·Xizhi Liu·Hongbin Zhao
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Abstract

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--Lidick\'{y}~[Surveys in combinatorics 2022, 21-63] in a stronger form.

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Levente Bodnár, Jinghua Deng, Jianfeng Hou, Xizhi Liu, Hongbin Zhao. 2025-07-01. Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm. https://arxiv.org/abs/2507.00812

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