Separating hypergraph Turán densities
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913684344799232 |
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| author | Liu, Hong Schülke, Bjarne Wang, Shuaichao Yang, Haotian Zhang, Yixiao |
| author_facet | Liu, Hong Schülke, Bjarne Wang, Shuaichao Yang, Haotian Zhang, Yixiao |
| contents | Determining the Turán densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Turán posed this problem in 1941, $π(K_{\ell}^{(k)})$ remains unknown for all $\ell>k\geq 3$. Prior to this work, it was not even known whether $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$ holds for general $\ell$ and $k$, and the best-known bounds on $π(K_{\ell}^{(k)})$ are far from implying anything close to this. We prove that $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$, for all $\ell>k\geq 3$, and provide a general criterion to distinguish the Turán densities of two hypergraphs. As a corollary, we obtain that $π(K_{k+1}^{(k)})<π(K_{k+2}^{(k)-})$, for all $k\geq 3$. For $k=3$, this was previously proved by Markström, answering a question by Erdős. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Separating hypergraph Turán densities Liu, Hong Schülke, Bjarne Wang, Shuaichao Yang, Haotian Zhang, Yixiao Combinatorics 05C65, 05C35, 05C42 Determining the Turán densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Turán posed this problem in 1941, $π(K_{\ell}^{(k)})$ remains unknown for all $\ell>k\geq 3$. Prior to this work, it was not even known whether $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$ holds for general $\ell$ and $k$, and the best-known bounds on $π(K_{\ell}^{(k)})$ are far from implying anything close to this. We prove that $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$, for all $\ell>k\geq 3$, and provide a general criterion to distinguish the Turán densities of two hypergraphs. As a corollary, we obtain that $π(K_{k+1}^{(k)})<π(K_{k+2}^{(k)-})$, for all $k\geq 3$. For $k=3$, this was previously proved by Markström, answering a question by Erdős. |
| title | Separating hypergraph Turán densities |
| topic | Combinatorics 05C65, 05C35, 05C42 |
| url | https://arxiv.org/abs/2410.08921 |