# Chromatic number of duals of uniform hypergraphs with large edges

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 $$\chi(H^\partial) = 2$$?

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 $$\chi(H^\partial)=2$$.