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