An improved hypergraph Mantel's Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916660298907648 |
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| author | Iľkovič, Daniel Yan, Jun |
| author_facet | Iľkovič, Daniel Yan, Jun |
| contents | In a recent paper, Chao and Yu used an entropy method to show that the Turán density of a certain family $\mathcal{F}$ of $\lfloor r/2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!/r^r$. Later, Liu determined for large $n$ the exact Turán number $\text{ex}(n,\mathcal{F})$ of this family, and showed that the unique extremal graph is the balanced complete $r$-partite $r$-uniform hypergraph. These two results together can be viewed as a hypergraph version of Mantel's Theorem. In this paper, building on their methods, we improve both of these results by showing that they still hold with a subfamily $\mathcal{F}'\subset\mathcal{F}$ of size $\lceil r/e\rceil$ in place of $\mathcal{F}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improved hypergraph Mantel's Theorem Iľkovič, Daniel Yan, Jun Combinatorics 05C35, 05C65, 94A17 In a recent paper, Chao and Yu used an entropy method to show that the Turán density of a certain family $\mathcal{F}$ of $\lfloor r/2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!/r^r$. Later, Liu determined for large $n$ the exact Turán number $\text{ex}(n,\mathcal{F})$ of this family, and showed that the unique extremal graph is the balanced complete $r$-partite $r$-uniform hypergraph. These two results together can be viewed as a hypergraph version of Mantel's Theorem. In this paper, building on their methods, we improve both of these results by showing that they still hold with a subfamily $\mathcal{F}'\subset\mathcal{F}$ of size $\lceil r/e\rceil$ in place of $\mathcal{F}$. |
| title | An improved hypergraph Mantel's Theorem |
| topic | Combinatorics 05C35, 05C65, 94A17 |
| url | https://arxiv.org/abs/2503.14474 |