SCHWEITZER 2008 Problem 1

Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$ . Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a "unique" point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$ .

(translated by Miklós Maróti)