Papers › The Turán density of short tight cycles
The Turán density of short tight cycles
Levente Bodnár, Jared León, Xizhi Liu, Oleg Pikhurko
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The 3-uniform tight ℓ-cycle C_ℓ³ is the 3-graph on {1,…,ℓ} consisting of all ℓ consecutive triples in the cyclic order. Let 𝒞 be either the pair {C₄³, C₅³} or the single tight ℓ-cycle C_ℓ³ for some ℓ≥7 not divisible by 3. We show that the Tur\'an density of 𝒞, that is, the asymptotically maximal edge density of a large 𝒞-free 3-graph, is equal to 2√(3) - 3. We also establish the corresponding Erd\H{o}s-Simonovits-type stability result, informally stating that all almost maximum 𝒞-free graphs are close in the edit distance to a 2-part recursive construction. This extends the earlier analogous results of Kam\v{c}ev-Letzter-Pokrovskiy ["The Tur\'an density of tight cycles in three-uniform hypergraphs", Int. Math. Res. Not. 6 (2024), 4804-4841] that apply for sufficiently large ℓ only. Additionally, we prove a finer structural result that allows us to determine the maximum number of edges in a {C₄³, C₅³}-free 3-graph with a given number of vertices up to an additive O(1) error term.
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