For any set $X$ and cardinal $\kappa$, we denote by $[X]^\kappa$ the collection of subsets of $X$ having cardinality $\kappa$.

If $H=(V,E)$ is a hypergraph, and $\kappa$ is a cardinal, we say that a map $c:V\to \kappa$ is a *coloring* if for every $e\in E$ with $|e|\geq 2$ the restriction $c|_e$ is non-constant. By $\chi(H)$ we denote the minimum cardinal such that there is a coloring $c:V\to \kappa$.

Let cardinals $\kappa\geq \omega$ and $\alpha,\beta$ with $2< \alpha, \beta \leq \kappa$ be given.

Is there necessarily $E \subseteq [\kappa]^\beta$ such that $\chi(\kappa,E) = \alpha$? (There is an easy positive answer for either $\alpha$ or $\beta$ equal to $2$.)