In this paper we investigate density conditions for finding a complete r-uniform hypergraph K-(r+1((r)) on r + 1 vertices in an (r + 1)-partite r-uniform hypergraph G. First we prove an optimal condition in terms of the densities of the (r + 1) induced r-partite subgraphs of G. Second, we prove a version of this result where we assume that r-tuples of vertices in G have their neighbours evenly distributed in G. Third, we also prove a counting result for the minimum number of copies of K-r+1((r)) when G satisfies our density bound, and present some open problems.
A striking difference between the graph, r = 2, and the hypergraph, r >= 3, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the r-partite parts needed to ensure the existence of a complete r-graph with (r + 1) vertices is equal to the golden ratio τ = 0.618 ... for r = 2, while it is r/r+1 for r >= 3.