The codegree Turán density of tight cycles
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908736770015232 |
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| author | Ma, Jie Rong, Mingyuan |
| author_facet | Ma, Jie Rong, Mingyuan |
| contents | The codegree Turán density $γ(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $γ\ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained in at least $(γ+o(1))n$ edges, contains a copy of $F$. A recent result of Piga, Sanhueza-Matamala, and Schacht determines that $γ(C_{\ell}^3)=\frac13$ for every $3$-uniform tight cycle $C_\ell^3$ of length $\ell$, where $\ell \ge \ell_0$ and $\ell$ is not divisible by $3$. In this paper, we investigate the codegree Turán density of $k$-uniform tight cycles $C_\ell^k$. We establish improved upper and lower bounds on $γ(C_{\ell}^k)$ for general $\ell$ not divisible by $k$. These results yield the following consequences:
1). For any prime $k \ge 3$, we show that $γ(C_{\ell}^k)=\frac13$ for all sufficiently large $\ell$ not divisible by $k$, generalizing the above theorem of Piga et al.
2). For all $k \ge 3$, we determine the exact value of $γ(C_{\ell}^k)$ for integers $\ell$ not divisible by $k$ in a set of (natural) density at least $\frac{φ(k)}{k}$, where $φ(\cdot)$ denotes Euler's totient function.
3). We give a complete answer to a question of Han, Lo, and Sanhueza-Matamala concerning the tightness of their construction for $γ(C_{\ell}^k)$.
Moreover, our results also determine the codegree Turán density of $C_\ell^{k-}$, that is, the $k$-uniform tight cycle of length $\ell$ with one edge removed, for a new set of integers $\ell$ of positive density for every $k \ge 3$. Our upper bound result is based on a structural characterization of $C_{\ell}^k$-free $k$-uniform hypergraphs with high minimum codegree, while the lower bounds are derived from a novel construction model, coupled with the arithmetic properties of the integers $k$ and $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23011 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The codegree Turán density of tight cycles Ma, Jie Rong, Mingyuan Combinatorics The codegree Turán density $γ(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $γ\ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained in at least $(γ+o(1))n$ edges, contains a copy of $F$. A recent result of Piga, Sanhueza-Matamala, and Schacht determines that $γ(C_{\ell}^3)=\frac13$ for every $3$-uniform tight cycle $C_\ell^3$ of length $\ell$, where $\ell \ge \ell_0$ and $\ell$ is not divisible by $3$. In this paper, we investigate the codegree Turán density of $k$-uniform tight cycles $C_\ell^k$. We establish improved upper and lower bounds on $γ(C_{\ell}^k)$ for general $\ell$ not divisible by $k$. These results yield the following consequences: 1). For any prime $k \ge 3$, we show that $γ(C_{\ell}^k)=\frac13$ for all sufficiently large $\ell$ not divisible by $k$, generalizing the above theorem of Piga et al. 2). For all $k \ge 3$, we determine the exact value of $γ(C_{\ell}^k)$ for integers $\ell$ not divisible by $k$ in a set of (natural) density at least $\frac{φ(k)}{k}$, where $φ(\cdot)$ denotes Euler's totient function. 3). We give a complete answer to a question of Han, Lo, and Sanhueza-Matamala concerning the tightness of their construction for $γ(C_{\ell}^k)$. Moreover, our results also determine the codegree Turán density of $C_\ell^{k-}$, that is, the $k$-uniform tight cycle of length $\ell$ with one edge removed, for a new set of integers $\ell$ of positive density for every $k \ge 3$. Our upper bound result is based on a structural characterization of $C_{\ell}^k$-free $k$-uniform hypergraphs with high minimum codegree, while the lower bounds are derived from a novel construction model, coupled with the arithmetic properties of the integers $k$ and $\ell$. |
| title | The codegree Turán density of tight cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.23011 |