If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/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 $\chi(H^\partial) = \lambda$?



Chromatic number and taking duals of hypergraphs
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 $H=K\_3\cup K\_\lambda^\partial$.


