Let $H=(V,E)$ be a hypergraph, let $n\in\mathbb{N}$. A *vertex coloring* is a map $c: V\to \{1,\ldots, n\}$ such that for $v\neq w \in V$ we have $c(v)\neq c(w)$ whenever there is $e\in E$ such that $v, w\in e$. We call the least $m\in\mathbb{N}$ such that there is a coloring map from $V$ to $\{1,\ldots,m\}$ the *chromatic number* $\chi(H)$ of $H=(V,E)$.

If $|e| = n$ for all $e\in E$ we say that $H=(V,E)$ is $n$-*regular*.

Let $n\in\mathbb{N}$. What is the maximum chromatic number that an $n$-regular hypergraph $H=(V,E)$ with $|E|=n$ can have?