Complete r-partite subgraphs of dense r-graphs

Let r ≥ 3 and (lnn)−1/(r−1) ≤ α ≤ r−3. We show that: Every r-uniform graph on n vertices with at least αnr/r! edges contains a complete r-partite graph with r − 1 parts of size ⌊ α (lnn)1/(r−1) ⌋ and one part of size ⌈ n1−α r−2 ⌉ . This result follows from a more general digraph version: Let U1, . . . , Ur be sets of size n, and M ⊂ U1 × · · · × Ur satisfy |M | ≥ αnr. If the integers s1, . . . , sr−1 satisfy 1 ≤ s1 · · · sr−1 ≤ ⌊ αr−1 lnn ⌋ , then there exists V1 × · · · × Vr ⊂ M, such that Vi ⊂ Ui and |Vi| = si for 1 ≤ i < r, and |Vr| > n1−α r−2 .