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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2305.11749 |
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| _version_ | 1866918073754189824 |
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| author | Li, Hao Lin, Hao Wang, Guanghui Zhou, Wenling |
| author_facet | Li, Hao Lin, Hao Wang, Guanghui Zhou, Wenling |
| contents | The uniform Turán density $π_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $π_{1}(F)$ for given hypergraphs $F$ was suggested by Erdős and Sós in 1980s. In particular, they raised the questions of determining $π_{1}(K_4^{(3)-})$ and $π_{1}(K_4^{(3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{(3)-}$, there are very few hypergraphs whose uniform Turán density has been determined.
In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $π_{1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Turán density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F^{\star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Turán density is also $1/4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11749 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hypergraphs with a quarter uniform Turán density Li, Hao Lin, Hao Wang, Guanghui Zhou, Wenling Combinatorics The uniform Turán density $π_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $π_{1}(F)$ for given hypergraphs $F$ was suggested by Erdős and Sós in 1980s. In particular, they raised the questions of determining $π_{1}(K_4^{(3)-})$ and $π_{1}(K_4^{(3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{(3)-}$, there are very few hypergraphs whose uniform Turán density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $π_{1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Turán density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F^{\star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Turán density is also $1/4$. |
| title | Hypergraphs with a quarter uniform Turán density |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.11749 |