Codegree threshold for tiling $k$-graphs with two edges sharing exactly $\ell$ vertices

Abstract

Given integer $k$ and a $k$-graph $F$, let $t_{k-1}(n,F)$ be the minimum integer $t$ such that every $k$-graph $H$ on $n$ vertices with codegree at least $t$ contains an $F$-factor. For integers $k\ge 3$ and $0\le \ell \le k-1$, let ${\mathcal{Y}}_{k,\ell }$ be a $k$-graph with two edges that shares exactly $\ell$ vertices. Han and Zhao (JCTA, 2015) asked the following question: For all $k\ge 3$, $0\le \ell \le k-1$ and sufficiently large $n$ divisible by $2k-\ell$, determine the exact value of $t_{k-1}(n,{\mathcal{Y}}_{k,\ell })$. In this paper, we show that $t_{k-1}(n,{\mathcal{Y}}_{k,\ell })=\frac{n}{2k-\ell }$ for $k\ge 3$ and $1\le \ell \le k-2$, combining with two previously known results of Rödl, Ruciński and Szemerédi {(JCTA, 2009)} and Gao, Han and Zhao (arXiv, 2016), the question of Han and Zhao is solved completely.

10 pages