A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A *coloring* is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal, such that for every $e\in E$ with $|e|\geq 2$ the restriction $c|_e$ is non-constant.

**Question.** Is every hypergraph $H=(V,E)$ with $|V|\geq \omega$ and $|E| = |V|$ and $|e| = |V|$ for all $e\in E$ $2$-colorable?

*Motivation of question.* If we take $V= \omega$ and $E$ to be the collection of computable subsets of $\omega$, then the resulting hypergraph is $2$-colorable - and there are even "balanced" colorings of $\omega$, also referred to as computationally random bitstreams.