F-factors in Quasi-random Hypergraphs

Given (Formula presented.) and two (Formula presented.) -graphs ((Formula presented.) -uniform hypergraphs) (Formula presented.) and (Formula presented.), an (Formula presented.) -factor in (Formula presented.) is a set of vertex-disjoint copies of (Formula presented.) that together cover the vertex set of (Formula presented.). Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the (Formula presented.) -factor problem in quasi-random (Formula presented.) -graphs with minimum degree (Formula presented.). They posed the problem of characterizing the (Formula presented.) -graphs (Formula presented.) such that every sufficiently large quasi-random (Formula presented.) -graph with constant edge density and minimum degree (Formula presented.) contains an (Formula presented.) -factor, and, in particular, they showed that all linear (Formula presented.) -graphs satisfy this property. In this paper we prove a general theorem on (Formula presented.) -factors which reduces the (Formula presented.) -factor problem of Lenz and Mubayi to a natural sub-problem, that is, the (Formula presented.) -cover problem. By using this result, we answer the question of Lenz and Mubayi for those (Formula presented.) which are (Formula presented.) -partite (Formula presented.) -graphs, and for all 3-graphs (Formula presented.), 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 (Formula presented.) -graphs.