Large chromatic number in hypergraphs with large edges

Question

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 $\chi(\kappa,E) = \kappa$?

Answer 1

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.