Let $H = (V,E)$ be a hypergraph. For $v\in V$ we set $E_v = \{e\in E: v\in E\}$. The dual of $H$ is defined by $H^* =(E, V^*)$ is, where $V^* = \{E_v:v\in V\}$.
If $\kappa>0$ is a cardinal, a map $c:V\to \kappa$ is a coloring of $H=(V,E)$ if the restriction $c|_e:e\to \kappa$ is non-constant whenever $|e|>1$. The smallest cardinal $\kappa>0$ such that there is a coloring $c:V\to\kappa$ is said to be the chromatic number $\chi(H)$ of $H$.
If $\kappa,\lambda >1$ are cardinals, is there necessarily a hypergraph $H$ with $\chi(H) = \kappa$ and $\chi(H^*) = \lambda$?
