If $n\in\mathbb{N}$ is a non-negative integer, we consider it as a cardinal, so $n = \{0, \ldots, n-1\}$. If $X$ is a set, and $\kappa$ is a cardinal, we let $[X]^\kappa$ be the collection of subsets of $X$ having cardinality $\kappa$.


If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa\neq \emptyset$ is a cardinal, a map $c:V\to \kappa$ is said to be a *coloring* if the restriction $c\restriction\_e$ is non-constant whenever $e\in E$ has more than $1$ element. The smallest cardinal $\kappa$ such that there is a coloring map $c:V\to \kappa$ is said to be the *chromatic number* of $H=(V,E)$, and we denote it by $\chi(H)$.


**Question.** Given integers $n\geq k \geq 3$, what is the value of $\chi(n,[n]^k)$ in terms of $n$ and $k$?



Chromatic number of $(n, [n]^k)$
It is $\lceil \frac{n}{k-1}\rceil$, simply since every color class must contain at most $k-1$ elements.


