Tilings in quasi-random $k$-partite hypergraphs

TL;DR

The paper establishes a sharp density threshold for embedding balanced $k$-partite $k$-graph factors in $(p, ext{μ})$-dense, quasi-random host hypergraphs under a vanishing partite minimum codegree condition, proving that $p>1/2$ suffices when each part has size $n$ and the factor $F$ has $m$ vertices per part. It introduces a lattice-based absorption framework with two key absorbing lemmas tailored to different density regimes, and combines absorption with almost-tiling arguments to obtain perfect tilings. The work proves optimality of the $p>1/2$ threshold via probabilistic constructions and shows that, for $p>1/2$, the partite minimum codegree can vanish while still guaranteeing all fixed $k$-partite $k$-graph factors, while also delivering an asymptotic $n/2$ codegree threshold without quasi-randomness. Methodologically, the paper leverages the weak hypergraph regularity lemma, cluster hypergraphs, and reachability/closure tools to bridge local abundance of $F$-copies to global tiling. Overall, it advances the understanding of tilings in multipartite hypergraphs under quasi-randomness by combining probabilistic constructions, absorption, and regularity techniques to achieve tight thresholds and robust tiling results.

Abstract

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor problems in quasi-random $k$-graphs with a minimum degree condition. Recently, Ding, Han, Sun, Wang and Zhou gave the density threshold for having all $3$-partite $3$-graphs factors in quasi-random $3$-graphs with vanishing minimum codegree condition $Ω(n)$. In this paper, we consider embedding factors when the host $k$-graph is $k$-partite and quasi-random with partite minimum codegree condition. We prove that if $p>1/2$ and $F$ is a $k$-partite $k$-graph with each part having $m$ vertices, then for $n$ large enough and $m\mid n$, any $p$-dense $k$-partite $k$-graph with each part having $n$ vertices and partite minimum codegree condition $Ω(n)$ contains an $F$-factor. We also present a construction showing that $1/2$ is best possible. Furthermore, for $1\leq \ell \leq k-2$, by constructing a sequence of $p$-dense $k$-partite $k$-graphs with partite minimum $\ell$-degree $Ω(n^{k-\ell})$ having no $K_k(m)$-factor, we show that the partite minimum codegree constraint can not be replaced by other partite minimum degree conditions. On the other hand, we prove that $n/2$ is the asymptotic partite minimum codegree threshold for having all fixed $k$-partite $k$-graph factors in sufficiently large host $k$-partite $k$-graphs even without quasi-randomness.

Theorems & Definitions (36)

• Definition 1.1: ($p,\mu$)-denseness
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• proof : Proof of Theorem \ref{['cons1']}
• proof : Proof of Theorem \ref{['cons2']}
• Lemma 3.1: Absorbing Lemma I
• Lemma 3.2: Absorbing Lemma II
• Lemma 3.3