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](https://en.wikipedia.org/wiki/Recursive_set) 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](https://mathoverflow.net/questions/335181/computationally-random-bitstreams-and-normalcy).



Are hypergraphs $H=(V,E)$ with $|E|=|V|$ $2$-colorable?
This is essentially done by the Bernstein set construction: if one has $\kappa$ many sets each of size $\kappa$, then order them into ordinal $\kappa$ and recursively choose 2 points from each, so that all these points are distinct. That is, we have $x\_\alpha,y\_\alpha\in A\_\alpha$ with all $x\_\alpha,y\_\alpha$ distinct. At the end, color each $x\_\alpha$ red, each $y\_\alpha$ green. 


