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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 thedensities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we provea 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 acounting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$satisfies our density bound, and present some open problems. A strikingdifference between the graph, $r=2$, and the hypergraph, $ r \geq 3 $, cases isthat in the first case both the existence threshold and the counting functionare non-linear in the involved densities, whereas for hypergraphs they aregiven by a linear function. Also, the smallest density of the $r$-partite partsneeded to ensure the existence of a complete $r$-graph with $(r+1)$ vertices isequal to the golden ratio $\tau=0.618\ldots$ for $r=2$, while it is$\frac{r}{r+1}$for $r\geq3$.

Partite Tur\'an-densities for complete $r-$uniform hypergraphs on $r+1$ vertices