$F$-factors in hypergraphs via absorption

Abstract

Given integers $ n \ge k >l \ge 1 $ and a $k$-graph $F$ with $|V(F)|$ divisible by $n$, define $t_l^k(n,F)$ to be the smallest integer $d$ such that every $k$-graph $H$ of order $n$ with minimum $l$-degree $\delta_l(H) \ge d $ contains an $F$-factor. A classical theorem of Hajnal and Szemer\'{e}di implies that $t^2_1(n,K_t) = (1-1/t)n$ for integers $t$. For $k \ge 3$, $t^k_{k-1}(n,K_k^k)$ (the $\delta_{k-1}(H)$ threshold for perfect matchings) has been determined by K\"{u}hn and Osthus (asymptotically) and R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di (exactly) for large $n$. In this paper, we generalise the absorption technique of R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di to $F$-factors. We determine the asymptotic values of $t^k_1(n,K_k^k(m))$ for $k = 3,4$ and $m \ge 1$. In addition, we show that for $t>k = 3$ and $\gamma >0$, $ t^3_{2}(n,K_t^3) \le (1- \frac{2}{t^2-3t+4} + \gamma) n$ provided $n$ is large and $t | n$. We also bound $t^3_{2}(n,K_t^3)$ from below. In particular, we deduce that $t^3_2(n,K_4^3) = (3/4+o(1))n$ answering a question of Pikhurko. In addition, we prove that $t^k_{k-1}(n,K_t^k) \le (1- \binom{t-1}{k-1}^{-1} + \gamma)n$ for $\gamma >0$, $k \ge 6$ and $t \ge (3+ \sqrt5)k/2$ provided $n$ is large and $t | n$.