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arXiv · 2412.21011

The Tur\'an density of the tight 5-cycle minus one edge

Levente Bodnár·Jared León·Xizhi Liu·Oleg Pikhurko
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Abstract

Let the tight $\ell$-cycle minus one edge $C_\ell^{3-}$ be the $3$-graph on $\{1,\dots,\ell\}$ consisting of $\ell-1$ consecutive triples in the cyclic order. We show that, for every $\ell\ge 5$ not divisible by $3$, the Tur\'an density of $C_{\ell}^{3-}$ is $1/4$ and also prove some finer structure results. This proves a conjecture of Mubayi--Sudakov--Pikhurko from 2011 and extends the results of Balogh--Luo [Combinatorica 44 (2024) 949--976] who established analogous claims for all sufficiently large $\ell$. Results similar to ours were independently obtained by Lidick\'y--Mattes--Pfender [arXiv:2409.14257].

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Levente Bodnár, Jared León, Xizhi Liu, Oleg Pikhurko. 2024-12-30. The Tur\'an density of the tight 5-cycle minus one edge. https://arxiv.org/abs/2412.21011

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