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?



Forbidden Tripartite Graphs
The [Erdos-Stone theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem) is the reference you are looking for.


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](http://jlms.oxfordjournals.org/content/s2-23/2/207), for the complete tripartite graph $G=K\_3(t)$, which has $t$ vertices in each part, the upper bound can be written as $n^2/4 + n^{2-1/500t}$.


