Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* of $X$ if $\bigcup {\cal C} = X$. A subset $S\subseteq X$ is a *choice set* for ${\cal C}$ if $|S\cap K| = 1$ for all $K\in {\cal C}$.

Suppose $X$ is infinite and ${\cal C}$ is a cover of $X$ with the following properties:

- if $K\in {\cal C}$, then $|K| = |X|$, and
- for $K\neq L \in {\cal C}$ we have $|K\cap L| <|X|$.

Is there a choice set for ${\cal C}$?