Papers › The Turán density of the tight 5-cycle minus one edge
The Turán density of the tight 5-cycle minus one edge
Levente Bodnár, Jared León, Xizhi Liu, Oleg Pikhurko
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Let the tight ℓ-cycle minus one edge C_ℓ³⁻ be the 3-graph on {1,…,ℓ} consisting of ℓ-1 consecutive triples in the cyclic order. We show that, for every ℓ≥5 not divisible by 3, the Tur\'an density of C_ℓ³⁻ 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 ℓ. Results similar to ours were independently obtained by Lidick\'y--Mattes--Pfender [arXiv:2409.14257].
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