Turán density of stars in uniformly dense hypergraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917467167653888 |
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| author | Lin, Hao Zhou, Wenling |
| author_facet | Lin, Hao Zhou, Wenling |
| contents | A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,μ,1)$-\emph{dense} if for any subsets $X,Y,Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Z$ such that $\{x,y,z\}$ is an edge of $H$ is at least $d|X||Y||Z|-μ|V|^3$. The \emph{$k$-star} $S_k$ is the $3$-graph with a center vertex and $k$ distinct leaf vertices, whose edge set consists of all triples containing the center and two distinct leaves. Restricting to $dot$-dense $3$-graphs, determining the \emph{$1$-uniform Turán density} $π_1(S_k)$ of $S_k$ for $k\ge 4$ was proposed by Schacht in ICM 2022. In particular, Reiher, Rödl and Schacht gave a palette construction showing that $π_1(S_k)\ge \frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 3$, and also proved that $π_1(S_3)=1/4$. Lamaison and Wu later showed that this palette construction is optimal for $k\ge 48$.
In this paper, we improve the results of Lamaison and Wu by proving that \[ π_1(S_k)=\frac{k^2-5k+7}{(k-1)^2} \qquad\text{for all } k\ge 9. \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12576 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turán density of stars in uniformly dense hypergraphs Lin, Hao Zhou, Wenling Combinatorics A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,μ,1)$-\emph{dense} if for any subsets $X,Y,Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Z$ such that $\{x,y,z\}$ is an edge of $H$ is at least $d|X||Y||Z|-μ|V|^3$. The \emph{$k$-star} $S_k$ is the $3$-graph with a center vertex and $k$ distinct leaf vertices, whose edge set consists of all triples containing the center and two distinct leaves. Restricting to $dot$-dense $3$-graphs, determining the \emph{$1$-uniform Turán density} $π_1(S_k)$ of $S_k$ for $k\ge 4$ was proposed by Schacht in ICM 2022. In particular, Reiher, Rödl and Schacht gave a palette construction showing that $π_1(S_k)\ge \frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 3$, and also proved that $π_1(S_3)=1/4$. Lamaison and Wu later showed that this palette construction is optimal for $k\ge 48$. In this paper, we improve the results of Lamaison and Wu by proving that \[ π_1(S_k)=\frac{k^2-5k+7}{(k-1)^2} \qquad\text{for all } k\ge 9. \] |
| title | Turán density of stars in uniformly dense hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.12576 |