# Chromatic number and taking duals of hypergraphs

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 $$\chi(H^\partial) = \lambda$$?

If I understand correctly, $$\chi(H^\partial)$$ is the least number of colours which suffice to colour the edges of $$H$$ so that each vertex is incident with edges of at least two different colours. And it seems to me that $$H$$ is isomorphic to $$(H^\partial)^\partial$$ provided that, for any vertices $$x,y\in V(H)$$, there is an edge $$e\in E(H)$$ such that $$|e\cap\{x,y\}|=1$$. I believe the answer is affirmative for all $$\kappa,\lambda\ge2$$. Each of the example hypergraphs is either an ordinary graph, or the dual of a graph, or the disjoint union of a graph and the dual of a graph.

If $$\kappa\ge4$$ and $$\lambda=2$$, let $$H=K_\kappa$$ (the complete graph of order $$\kappa$$).

If $$\kappa=2$$ and $$\lambda\ge4$$, let $$H=K_\lambda^\partial$$.

If $$\kappa\ge4$$ and $$\lambda\ge4$$, let $$H=K_\kappa\cup K_\lambda^\partial$$ (disjoint union).

If $$\kappa=2$$ and $$\lambda=2$$, let $$H=C_4$$.

If $$\kappa=3$$ and $$\lambda=3$$, let $$H=K_3$$.

If $$\kappa=3$$ and $$\lambda=2$$, let $$H=K_4-e$$.

If $$\kappa=2$$ and $$\lambda=3$$, let $$H=(K_4-e)^\partial$$.

If $$\kappa\ge4$$ and $$\lambda=3$$, let $$H=K_\kappa\cup K_3$$.

If $$\kappa=3$$ and $$\lambda\ge4$$, let $$H=K_3\cup K_\lambda^\partial$$.