These are known as *Perfect Hash Families*. The notation $PHF(N;n,v,k)$ denotes a set $\mathcal H$ (hash functions) with $|\mathcal H|=N$, of $v$-colorings of the base $[n]$, such that for any $X\subset [n]$ with $|X|=k$, there exists at least one hash function which assigns it different colors. Let's denote by $f(n,v,k)$ the smallest $N$ for which a $PHF(N;n,v,k)$ exists. In this notation, you are interested in $f(n,k,k)$.

M. L. Fredman and J. Komlós prove in the paper "On the Size of Separating Systems and Families of Perfect Hash Functions" the following bounds:

$$\frac{v^{k-1}}{(v)_{k-1}}\cdot\frac{\log n}{\log(v-k+2)}\le f(n,v,k)\le \frac{k\log n}{\log\left(1-\frac{(v)_k}{v^k}\right)}$$
where $(v)_k=v(v-1)\cdots (v-k+1)$ is the falling factorial. This bound was improved by J. Körner and K. Marton in "New Bounds for Perfect Hashing via Information Theory" and there is a large literature making sharper bounds for particular families of parameters. Here is a large table of known parameters of realizable perfect hash families maintained by Ryan Dougherty.