$F$-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 Szemerédi in (Proof of a Conjecture of P. Erdős, pp. 601–623, 1969) 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 Kühn and Osthus in (J Graph Theory 51(4):269–280, 2006) (asymptotically) and Rödl et al. in (J Combin Theory Ser A 116(3):613–636, 2009) (exactly) for large $n$. In this paper, we generalise the absorption technique of Rödl et al. in (J Combin Theory Ser A 116(3):613–636, 2009) 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 (Graphs Combin 24(4):391–404, 2008). In addition, we prove that $t^k_{k-1}(n,K_t^k) \le (1- \left( {\begin{array}{c}t-1\\ k-1\end{array}}\right) ^{-1} + \gamma )n$ for $\gamma >0, k \ge 6$ and $t \ge (3+ \sqrt{5})k/2$ provided $n$ is large and $t | n$.