I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices and no $K_{s,t}$ subgraph. However, I was unable to find anything about the tripartite equivalent. Is anything known about the number of edges in an $n$ vertex graph that guarantee $K_{s,t,u}$ as a subgraph? Could someone suggest a reference?

For bipartite graphs, the number of edges that guarantee $K_{s,t}$ is of order $n^{2 - \frac{1}{\min(s,t)}}$. Is there a similar bound for tripartite graphs where the growth rate is determined by the size of the graph?