The $3$-uniform tight $\ell$-cycle $C_\ell^{ 3}$ is the $3$-graph on $\{ 1,\dots,\ell\}$ consisting of all $\ell$ consecutive triples in the cyclic order. Let $\mathcal{ C}$ be either the pair $\{ C_{ 4}^{ 3}, C_{ 5}^{ 3}\}$ or the single tight $\ell$-cycle $C_{ \ell}^{ 3}$ for some $\ell\ge 7$ not divisible by $3$. We show that the Tur\'an density of $\mathcal{ C}$, that is, the asymptotically maximal edge density of a large $\mathcal{ C}$-free $3$-graph, is equal to $2\sqrt{ 3} - 3$. We also establish the corresponding Erd\H{ o}s-Simonovits-type stability result, informally stating that all almost maximum $\mathcal{ C}$-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 $\ell$ only. Additionally, we prove a finer structural result that allows us to determine the maximum number of edges in a $\{ C_{ 4}^{ 3}, C_{ 5}^{ 3}\}$-free $3$-graph with a given number of vertices up to an additive $O(1)$ error term.
Categories: math.CO
