Tiling multipartite hypergraphs in Quasi-random Hypergraphs

TL;DR

This paper establishes the density threshold for the existence of $F$-factors in quasi-random 3-graphs, showing that above a certain density, all such graphs contain factors of specific 3-partite 3-graphs, while below it, counterexamples exist.

Contribution

It proves the exact density threshold for $F$-factors in quasi-random 3-graphs with minimum codegree conditions, extending previous work and identifying the limits of vertex degree conditions.

Findings

Density threshold for $F$-factors is $1/8$ in quasi-random 3-graphs.

Minimum codegree condition is necessary; vertex degree alone is insufficient.

Counterexamples exist below the threshold, showing the sharpness of the result.

Abstract

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $\Omega(n^2)$ and minimum codegree $\Omega(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense quasi-random $3$-graphs of order $n$ with minimum codegree $\Omega(n)$ and $f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$. Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree $\Omega (n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$.