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?



Coloring the uncountable Lebesgue-measurable sets of $\mathbb{R}$
It is continuum. The coloring with continuum many colors is clear (all points may have different color). Assume that we have $\kappa<c$ colors. Consider the Cantor set $K$. All its subsets are Lebesgue measurable. If some color contains uncountably many points from $K$, it constitutes a monochromatic edge. So, each color contains at most $\omega$ points from $K$, and totally $|K|\leqslant w\cdot \kappa=\kappa$, a contradiction.


