A *hypergraph* $H=(V,E)$ consists of a set $V$ and $E\subseteq {\mathcal P}(V)$, that is, $E$ consists of subsets of $V$ of arbitrary size. Obviously, a graph is a special kind of hypergraph.

Let $H=(V,E)$ be a hypergraph and $\kappa\neq \emptyset$ be a cardinal.
Then a map $c:V\to \kappa$ is said to be a *coloring* if for every $e\in E$ with $|e|\geq 2$ we have that the restriction $c\restriction_e$ is non-constant. The *chromatic
number* $\chi(H)$ of $H$ is the smallest cardinal $\kappa$ such there is a colouring $c:V\to \kappa$.

**Question.** What is $\chi(\mathbb{R}, E)$ where $E$ consists of all uncountable Lebesgue-measurable sets?