\(F\)-factors in hypergraphs via absorption

DOI10.1007/S00373-014-1410-8zbMATH Open1312.05099arXiv1105.3411OpenAlexW2171957589MaRDI QIDQ2345534

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Publication date: 22 May 2015

Published in: (Search for Journal in Brave)

Abstract: Given integers $ngek>lge1$ 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 $delt{a}_l(H)ged$ contains an $F$-factor. A classical theorem of Hajnal and Szemer'{e}di implies that $t_1^2(n,K_t)=(1-1/t)n$ for integers $t$. For $kge3$, $t_{k-1}^k(n,K_k^k)$ (the $delt{a}_{k-1}(H)$ 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 $t_1^k(n,K_k^k(m))$ for $k=3,4$ and $mge1$. In addition, we show that for $t>k=3$ and $gamma>0$, $t_2^3(n,K_t^3)le(1-frac2t^2-3t+4+gamma)n$ provided $n$ is large and $t|n$. We also bound $t_2^3(n,K_t^3)$ from below. In particular, we deduce that $t_2^3(n,K_4^3)=(3/4+o(1))n$ answering a question of Pikhurko. In addition, we prove that $$ for $gamma>0$, $kge6$ and $tge(3+sqrt5)k/2$ provided $n$ is large and $t|n$.

Full work available at URL: https://arxiv.org/abs/1105.3411

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