For any r-graph H, we consider the problem of finding a rainbow H-factor in an r-graph G with large minimum ℓ-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an H-factor.
IntroductionA fundamental question in Extremal Combinatorics is to determine conditions on a hypergraph G that guarantee an embedded copy of some other hypergraph H. The Turán problem for an r-graph H asks for the maximum number of edges in an r-graph G on n vertices; we usually think of H as fixed and n as large. For r = 2 (ordinary graphs) this problem is fairly well understood (except when H is bipartite), but for general r and general H we do not even have an asymptotic understanding of the Turán problem (see the survey [11]). For example, a classical theorem of Mantel determines the maximum number of edges in a triangle-free graph on n vertices (it is n 2 /4 ), but we do not know even asymptotically the maximum number of edges in a tetrahedron-free 3-graph on n vertices. On the other hand, if we seek to embed a spanning hypergraph then it is most natural to consider minimum degree conditions. Such questions are known as Dirac-type problems, after the classical theorem of Dirac that any graph on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle. There is a large literature on such problems for graphs and hypergraphs, surveyed in [15,17,22,28].One of these problems, finding hypergraph factors, will be our topic for the remainder of this paper. To describe it we introduce some notation and terminology. Let G be an r-graph on [n] = {1, . . . , n}. For any L ⊆ V (G) the degree of L in G is the number of edges of G containing L. The minimum ℓ-degree δ ℓ (G) is the minimum degree in G over all L ⊆ V (G) of size ℓ. Let H be an r-graph with |V (H)| = h | n. A partial H-factor F in G of size m is a set of m vertex-disjoint copies of H in G. If m = n/h we call F an H-factor. We let δ ℓ (H, n) be the minimum δ such that δ ℓ (G) > δn r−ℓ guarantees an H-factor in G. Then the asymptotic ℓ-degree threshold for H-factors is δ * ℓ (H) := lim inf m→∞ δ ℓ (H, mh) .