F$F$-factors in Quasi-random Hypergraphs

摘要

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

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