Are hypergraphs $H=(V,E)$ with $|E|=|V|$ $2$-colorable?

Question

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.

Answer 1

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.