Suppose that we have a complete graph $G$ of $n$ vertices. What is the minimum number of complete $k$-partite graph (subgraph of $G$) that covers all the complete graph of $k$ vertices of $V(G)$? Are there any results related to this problem? Any lower bounds or upper bounds?

To phrase it in another way. Suppose that there are $n$ students entering in an exam of some multiple choice questions, each has $k$ choices. After the exam, you find that among any $k$ students, there is (at least) a question such that these $k$ students answer differently. Then what is the minimum number of questions in this exam?

For example, when $k=2$, it is asking how many bipartite graph is need to cover a complete graph.