On the thresholds of degenerate hypergraphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929616603578368 |
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| author | Chen, Yu Han, Jie Luo, Haoran |
| author_facet | Chen, Yu Han, Jie Luo, Haoran |
| contents | An $n$-vertex $k$-uniform hypergraph $G$ is $(d,α)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have $e\left(G[U]\right)\le{d\left(|U|-1\right)-α}$. These hypergraphs include many natural graph classes, such as the degenerate hypergraphs, the planar graphs, and the power of cycles. In this paper, we consider the threshold of the emergence of a $(d,α)$-degenerate hypergraph with bounded maximum degree in the Erdős-Rényi model. We show that its threshold is at most $n^{-1/d}$, improving previous results of Riordan and Kelly-Müyesser-Pokrovskiy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18596 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the thresholds of degenerate hypergraphs Chen, Yu Han, Jie Luo, Haoran Combinatorics Probability 05C80, 60C05, 05C65 An $n$-vertex $k$-uniform hypergraph $G$ is $(d,α)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have $e\left(G[U]\right)\le{d\left(|U|-1\right)-α}$. These hypergraphs include many natural graph classes, such as the degenerate hypergraphs, the planar graphs, and the power of cycles. In this paper, we consider the threshold of the emergence of a $(d,α)$-degenerate hypergraph with bounded maximum degree in the Erdős-Rényi model. We show that its threshold is at most $n^{-1/d}$, improving previous results of Riordan and Kelly-Müyesser-Pokrovskiy. |
| title | On the thresholds of degenerate hypergraphs |
| topic | Combinatorics Probability 05C80, 60C05, 05C65 |
| url | https://arxiv.org/abs/2411.18596 |