In combinatorial terms, I was wondering how many complete $r$-partite $k$-uniform hypergraphs are needed to cover the edges of the complete $n$-vertex $k$-uniform hypergraph $\binom{[n]}{k}$. An $r$-partite $k$-uniform hypergraph is one where the vertices are partitioned into $r$ parts, with every edge (an edge is a set of $k$ vertices) having at most $1$ vertex from each part.
In non-combinatorial terms, for $n \ge r \ge t$, what is the minimum $t$ such that we can have $n$ vectors in $[r]^t$ with the property that for any subset of $k$ of the $n$ vectors, there is some coordinate where they have pairwise-distinct entries?
We have a lower bound of $t \ge \log_r n$, since otherwise there will be a pair of vectors (or vertices) that have the same coordinates (or are always in the same part). Random vectors give an upper bound of the form $t \le \log \binom{n}{k} / \log \left( \frac{r}{k^2} \right)$.
I am most interested in the case where $r = 2^k - 1$ and $n$ is large. In this case the above bounds give $t = \Omega \left( \frac{\log n}{k} \right)$ and $t = O \left( \log n \right)$. What is the correct dependence on $k$?