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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2507.12740 |
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| _version_ | 1866916847965700096 |
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| author | Han, Jie Hu, Jie Wei, Shunan Yang, Donglei |
| author_facet | Han, Jie Hu, Jie Wei, Shunan Yang, Donglei |
| contents | Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and $δ_{F,d}^T$ be the infimum of $δ>0$ such that for every $α>0$ and sufficiently large $n\in \mathbb{N}$, every $k$-graph system $\mathbf H=\{H_{1}, H_{2}, \dots ,H_{tn}\}$ on the same $sn$ vertices with $δ_d(H_i)\ge (δ+α)\binom{sn-d}{k-d}$, $i\in [tn]$ contains a transversal $F$-factor, that is, an $F$-factor consisting of exactly one edge from each $H_i$. In this paper we prove the following result. Let $\mathbf{H} =\{H_{1}, H_{2}, \dots ,H_{tn}\}$ be a $k$-graph system where each $H_{i}$ is an $sn$-vertex $k$-graph with $δ_d(H_i)\ge (δ_{F,d}^T+α)\binom{sn-d}{k-d}$. Then with high probability $\mathbf{H}(p) :=\{H_{1}(p), H_{2}(p), \dots ,H_{tn}(p)\}$ contains a transversal $F$-factor, where $H_i(p)$ is a random subhypergraph of $H_i$ and $p=Ω(n^{-1/d_1(F)-1}(\log n)^{1/t})$. This extends a recent result by Kelly, Müyesser and Pokrovskiy, and independently by Joos, Lang and Sanhueza-Matamala. Moreover, the assumption on $p$ is best possible up to a constant. Along the way, we also obtain a spread version of a result of Pikhurko on perfect matchings in $k$-partite $k$-graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transversal packings in families of percolated hypergraphs Han, Jie Hu, Jie Wei, Shunan Yang, Donglei Combinatorics Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and $δ_{F,d}^T$ be the infimum of $δ>0$ such that for every $α>0$ and sufficiently large $n\in \mathbb{N}$, every $k$-graph system $\mathbf H=\{H_{1}, H_{2}, \dots ,H_{tn}\}$ on the same $sn$ vertices with $δ_d(H_i)\ge (δ+α)\binom{sn-d}{k-d}$, $i\in [tn]$ contains a transversal $F$-factor, that is, an $F$-factor consisting of exactly one edge from each $H_i$. In this paper we prove the following result. Let $\mathbf{H} =\{H_{1}, H_{2}, \dots ,H_{tn}\}$ be a $k$-graph system where each $H_{i}$ is an $sn$-vertex $k$-graph with $δ_d(H_i)\ge (δ_{F,d}^T+α)\binom{sn-d}{k-d}$. Then with high probability $\mathbf{H}(p) :=\{H_{1}(p), H_{2}(p), \dots ,H_{tn}(p)\}$ contains a transversal $F$-factor, where $H_i(p)$ is a random subhypergraph of $H_i$ and $p=Ω(n^{-1/d_1(F)-1}(\log n)^{1/t})$. This extends a recent result by Kelly, Müyesser and Pokrovskiy, and independently by Joos, Lang and Sanhueza-Matamala. Moreover, the assumption on $p$ is best possible up to a constant. Along the way, we also obtain a spread version of a result of Pikhurko on perfect matchings in $k$-partite $k$-graphs. |
| title | Transversal packings in families of percolated hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.12740 |