Let $H=(V,E)$ be a hypergraph, and $\kappa$ be a cardinal. We say that a map $c:V \to \kappa$ is a *coloring* if the restriction $c\restriction_e$ is non-constant whenever $e\in E$ and $|e|\geq 2$. The smallest cardinal $\kappa$ such that there is a coloring $c:V\to\kappa$ is said to be the *chromatic number* of $H$, and we denote it by $\chi(H)$.

Let $\omega$ denote the set of non-negative integers. We say that a hypergraph $H=(\omega,E)$ is a *rainbow hypergraph* if every member of $E$ is finite, and if for every $n\in\omega\setminus\{0,1\}$ there is exactly one $e\in E$ with $|e|=n$ (that is, for all $n\in\omega\setminus\{0,1\}$ we have $|\{e\in E: |e| = n\}|=1$).

There are many rainbow hypergraphs with chromatic number $2$ (for instance, any hypergraph in which the edges are pairwise disjoint.)

**Question.** Given $k\in (\omega\cup\{\omega\})\setminus \{0,1\}$, is there a rainbow hypergraph $H=(\omega,E)$ with $\chi(H)=k$?