Transversal packings in families of percolated hypergraphs

TL;DR

This paper proves that under certain minimum degree conditions, a random subhypergraph system almost surely contains a transversal $F$-factor, extending recent results and establishing near-optimal probability bounds.

Contribution

It extends existing results on transversal packings in hypergraphs by establishing probabilistic thresholds for the existence of $F$-factors in random subhypergraphs under minimum degree conditions.

Findings

High probability existence of transversal $F$-factors in random subhypergraphs.

Optimality of the probability threshold up to a constant factor.

A spread version of a known result on perfect matchings in $k$-partite 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 $\delta_d(H_i)\ge (\delta+\alpha)\binom{sn-d}{k-d}$, $i\in [tn]$ 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 $\delta_d(H_i)\ge (\delta_{F,d}^T+\alpha)\binom{sn-d}{k-d}$. 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 $p=\Omega(n^{-1/d_1(F)-1}(\log n)^{1/t})$. This extends a recent result by Kelly, M\"{u}yesser 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.