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The Hypergraph Tur\'{a}n Densities of Tight Cycles Minus an Edge
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abstract
A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$, the extremal number is $ ex\left(C_\ell^-,n\right)=\tfrac1{24}n^3+O(n\ln n)=\left(\tfrac14+o(1)\right){n\choose 3}. $ We determine the extremal graph up to $O(n)$ edge edits.
Forward citations
Cited by 4 Pith papers
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Polynomial-to-exponential transition in 3-uniform Ramsey numbers
For every fixed s > 3, the 3-uniform Ramsey number r_3(s, g_3(s)+1; t) is at least 2^{c t^{2/3}} for some c > 0, settling the last open case of the 1972 Erdős-Hajnal conjecture.
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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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