# Large chromatic number in hypergraphs with large edges

Let $$H=(V,E)$$ be a hypergraph. If $$\kappa \neq \emptyset$$ is a cardinal, we call a map $$c:V\to \kappa$$ a coloring if for each $$e\in E$$ with $$|e|>1$$ the restriction $$c\restriction_e$$ is non-constant. The smallest cardinal $$\kappa > 0$$ such that there is a coloring map $$c:V\to \kappa$$ is said to be the chromatic number $$\chi(H)$$ of $$H$$.

Given an infinite cardinal $$\kappa$$, let $$[\kappa]^\kappa$$ denote the collection of subsets of $$\kappa$$ having cardinality $$\kappa$$.

A standard combinatorial argument shows that whenever $$E\subseteq [\kappa]^\kappa$$ has cardinality $$\kappa$$, then $$\chi(\kappa,E) = 2$$.

Question. Is it consistent with $${\sf ZFC}$$ that there is $$E \subseteq [\kappa]^\kappa$$ with $$|E|<2^\kappa$$ and $$\chi(\kappa,E) = \kappa$$?

For $$\kappa=\aleph_0$$ yes: there are (many) models with ultrafilters of character less than $$\mathfrak{c}$$. Let $$E\subseteq[\omega]^\omega$$ be a base for an ultrafilter, say $$|E|=\aleph_1<\mathfrak{c}$$. If $$f:\omega\to k$$ for some $$k<\omega$$ then $$f$$ is constant on a member of $$E$$. The identity map is a colouring of this graph.