Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/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 $\chi(H^\partial) = 2$?



Chromatic number of duals of uniform hypergraphs with large edges
**Theorem.** For any cardinals $\alpha,\beta\ge2$ there is an $\alpha$-uniform hypergraph $H$ with $\chi(H)=\beta$ and $\chi(H^\partial)=2$.


**Proof.** Let $V=\bigcup\_{\xi\in\beta}V\_\xi$ where the sets $V\_\xi$ are pairwise disjoint and $|V\_\xi|\gt\alpha\beta$. Let $E=\{e\in[V]^\alpha:|\{\xi\in\beta:e\cap V\_\xi\ne\varnothing\}|\ge2\}$.


Plainly $H=(V,E)$ is an $\alpha$-uniform hypergraph and $\chi(H)=\beta$. To see that $\chi(H^\partial)=2$ color each vertex red or blue so that each set $V\_\xi$ contains at least $\alpha$ vertices of each color. Then each vertex of $H$ is contained in both a monochromatic edge and a bichromatic edge, whence $\chi(H^\partial)=2$.


