Transversal packings in families of percolated hypergraphs

Abstract

Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and ${\delta }_{F,d}^T$ be the infimum of $\delta >0$ such that for every $\alpha >0$ 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 ${\delta }_d(H_i)\ge (\delta +\alpha )\left(\genfrac{}{}{0ex}{}{sn-d}{k-d}\right)$, $i\in [tn]$ 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 ${\delta }_d(H_i)\ge ({\delta }_{F,d}^T+\alpha )\left(\genfrac{}{}{0ex}{}{sn-d}{k-d}\right)$. 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 $p=\Omega (n^{-1/d_1(F)-1}(\log n{)}^{1/t})$. 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.

24 pages