Let $H=(V,E)$ be a hypergraph. If $\kappa>0$ is a cardinal, we say that $H$ is $\kappa$-*uniform* if $|e|=\kappa$ for all $e\in E$.

If $X$ is a non-empty set, then a map $c:V\to X$ is said to be a *colouring* if for every edge $e\in E$ with $|e|\geq 2$ the restriction $c\restriction_e: e\to X$ 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 $\alpha, \beta \geq 2$, is there an $\alpha$-uniform hypergraph $H=(V,E)$ with $\chi(H) = \beta$ and $\chi(H^\partial) = 2$?