F$F$‐factors in Quasi‐random Hypergraphs

Abstract

Given k ⩾ 2 $k\geqslant 2$ and two k $k$ -graphs ( k $k$ -uniform hypergraphs) F $F$ and H $H$ , an F $F$ -factor in H $H$ is a set of vertex-disjoint copies of F $F$ that together cover the vertex set of H $H$ . Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the F $F$ -factor problem in quasi-random k $k$ -graphs with minimum degree Ω ( n k − 1 ) $\Omega (n^{k-1})$ . They posed the problem of characterizing the k $k$ -graphs F $F$ such that every sufficiently large quasi-random k $k$ -graph with constant edge density and minimum degree Ω ( n k − 1 ) $\Omega (n^{k-1})$ contains an F $F$ -factor, and, in particular, they showed that all linear k $k$ -graphs satisfy this property. In this paper we prove a general theorem on F $F$ -factors which reduces the F $F$ -factor problem of Lenz and Mubayi to a natural sub-problem, that is, the F $F$ -cover problem. By using this result, we answer the question of Lenz and Mubayi for those F $F$ which are k $k$ -partite k $k$ -graphs, and for all 3-graphs F $F$ , separately. Our characterization result on 3-graphs is motivated by the recent work of Reiher, Rödl, and Schacht [J. Lond. Math. Soc., 2018] that classifies the 3-graphs with vanishing Turán density in quasi-random k $k$ -graphs.