The Turán density of short tight cycles

Abstract

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 Turán 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 Erdős-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 Kamčev-Letzter-Pokrovskiy ["The Turán 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.

28 pages, ancilliary files. arXiv admin note: substantial text overlap with arXiv:2412.21011