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The Tur\'an density of the tight 5-cycle minus one edge
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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].
Forward citations
Cited by 3 Pith papers
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Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm
The ℓ2-norm Turán density of the tight cycle minus one edge C_ℓ^{3-} is exactly 1/26 for every ℓ ≥ 5 with ℓ not divisible by 3, with a stability theorem.
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The Tur\'{a}n density of short tight cycles
The Turán density of every 3-uniform tight cycle of length ℓ≥7 with ℓ not divisible by 3, and of the pair {C4^3,C5^3}, is exactly 2√3−3.
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Exact Tur\'{a}n densities in triple systems
The authors prove exact Turán densities for three families of 3-graphs, including confirming Shi's conjecture that π(C_4^3, complement of F_5) equals 2*sqrt(3) - 3.
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