A Jump in the Codegree Turán Densities of Long Tight Cycles
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| Format: | Preprint |
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2026
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| author | Balogh, József Luo, Haoran Sankar, Maya |
| author_facet | Balogh, József Luo, Haoran Sankar, Maya |
| contents | We study the codegree Turán density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even, then the codegree Turán density of $\mathcal{C}_\ell^r$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is odd, then the codegree Turán density of $\mathcal{C}_\ell^r$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell$ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_18398 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Jump in the Codegree Turán Densities of Long Tight Cycles Balogh, József Luo, Haoran Sankar, Maya Combinatorics 05C35, 05C65 We study the codegree Turán density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even, then the codegree Turán density of $\mathcal{C}_\ell^r$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is odd, then the codegree Turán density of $\mathcal{C}_\ell^r$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell$ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph. |
| title | A Jump in the Codegree Turán Densities of Long Tight Cycles |
| topic | Combinatorics 05C35, 05C65 |
| url | https://arxiv.org/abs/2602.18398 |