The Turán density of short tight cycles
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915323578417152 |
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| author | Bodnár, Levente León, Jared Liu, Xizhi Pikhurko, Oleg |
| author_facet | Bodnár, Levente León, Jared Liu, Xizhi Pikhurko, Oleg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Turán density of short tight cycles Bodnár, Levente León, Jared Liu, Xizhi Pikhurko, Oleg Combinatorics 05C65, 05C35, 05D99 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. |
| title | The Turán density of short tight cycles |
| topic | Combinatorics 05C65, 05C35, 05D99 |
| url | https://arxiv.org/abs/2506.03223 |