Extremal constructions for apex partite hypergraphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914082543632384 |
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| author | Chen, Qiyuan Liu, Hong Ye, Ke |
| author_facet | Chen, Qiyuan Liu, Hong Ye, Ke |
| contents | We establish new lower bounds for the Turán and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices that share $\mathcal{ H}$ as a common link. We show that $ex(n,\mathcal{H}(k))=Ω_{\mathcal{ H}}(n^{d-\frac{1}{e(\mathcal{H})}})$ if $k$ is at least exponentially large in $e(\mathcal{H})$. Our bound is optimal for all Sidorenko hypergraphs $\mathcal{H}$ and verifies a conjecture of Lee for such hypergraphs.
In particular, for the complete $d$-partite $d$-uniform hypergraphs $\mathcal{K}^{(d)}_{s_1,\dots,s_d}$, our result implies that $ex(n,\mathcal{K}^{(d)}_{s_{1},\cdots,s_{d}})=Θ(n^{d-\frac{1}{s_{1}\cdots s_{d-1}}})$ if $s_{d}$ is at least exponentially large in terms of $s_{1}\cdots s_{d-1}$, improving the factorial condition of Pohoata and Zakharov and answering a question of Mubayi. Our method is a generalization of Bukh's random algebraic method [Duke Math.J. 2024] to hypergraphs, and extends to the sided Zarankiewicz problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal constructions for apex partite hypergraphs Chen, Qiyuan Liu, Hong Ye, Ke Combinatorics Commutative Algebra We establish new lower bounds for the Turán and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices that share $\mathcal{ H}$ as a common link. We show that $ex(n,\mathcal{H}(k))=Ω_{\mathcal{ H}}(n^{d-\frac{1}{e(\mathcal{H})}})$ if $k$ is at least exponentially large in $e(\mathcal{H})$. Our bound is optimal for all Sidorenko hypergraphs $\mathcal{H}$ and verifies a conjecture of Lee for such hypergraphs. In particular, for the complete $d$-partite $d$-uniform hypergraphs $\mathcal{K}^{(d)}_{s_1,\dots,s_d}$, our result implies that $ex(n,\mathcal{K}^{(d)}_{s_{1},\cdots,s_{d}})=Θ(n^{d-\frac{1}{s_{1}\cdots s_{d-1}}})$ if $s_{d}$ is at least exponentially large in terms of $s_{1}\cdots s_{d-1}$, improving the factorial condition of Pohoata and Zakharov and answering a question of Mubayi. Our method is a generalization of Bukh's random algebraic method [Duke Math.J. 2024] to hypergraphs, and extends to the sided Zarankiewicz problem. |
| title | Extremal constructions for apex partite hypergraphs |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2510.07997 |