Forbidden Tripartite Graphs

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?

For every tripartite graph $G$, the number of edges in an $n$-vertex graph that guarantees $G$ as a subgraph is $n^2/4 + o(n^2)$. It is easy to see that $n^2/4$ is a lower bound (for $n$ even), since the complete bipartite graph does not contain $G$ as a subgraph.
By a result of Chvatal and Szemeredi, for the complete tripartite graph $G=K_3(t)$, which has $t$ vertices in each part, the upper bound can be written as $n^2/4 + n^{2-1/500t}$.
• 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? – Halbort Aug 9 '15 at 3:29