If $H=(V,E)$ is a hypergraph and $\kappa\neq \emptyset$ is a cardinal, then a map $c:V\to \kappa$ is said to be a *colouring* if for every edge $e\in E$ with $|e|\geq 2$ the restriction $c\restriction_e: e\to \kappa$ is not constant. The smallest cardinal $\kappa$ for which there is a colouring $c:V \to \kappa$ is said to be the *chromatic number* of $H$ and is denoted by $\chi(H)$.

The *dual* of the hypergraph $H=(V,E)$ is $H^\partial = (E, E_V)$ where $$E_V = \big\{S\subseteq E: \text{there is }v_0\in V \text{ such that } S = \{e\in E: v_0\in e\}\big\}.$$

**Question.** Given cardinals $\kappa, \lambda \geq 2$, is there always a hypergraph $H$ with $\chi(H) = \kappa$ and $\chi(H^\partial) = \lambda$?