Tiling multipartite hypergraphs in quasi-random hypergraphs

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 covers the vertex set of H. Lenz and Mubayi studied the F-factor problems in quasi-random k-graphs with minimum degree Ω(n^k−1). In particular, they constructed a sequence of 1/8-dense quasi-random 3-graphs H(n) with minimum degree Ω(n²) and minimum codegree Ω(n) but with no K_2,2,2-factor. We prove that if p>1/8 and F is a 3-partite 3-graph with f vertices, then for sufficiently large n, all p-dense quasi-random 3-graphs of order n with minimum codegree Ω(n) and f|n have F-factors. That is, 1/8 is the density threshold for ensuring all 3-partite 3-graphs F-factors in quasi-random 3-graphs given a minimum codegree condition Ω(n). Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any p∈(0,1) and n≥n₀, there exist p-dense quasi-random 3-graphs of order n with minimum degree Ω(n²) having no K_2,2,2-factor. In particular, we study the optimal density threshold of F-factors for each 3-partite 3-graph F in quasi-random 3-graphs given a minimum codegree condition Ω(n). In addition, we also study F-factor problems for k-partite k-graphs F with stronger quasi-random assumption and minimum degree Ω(n^k−1).