Question 39
Let $n \gt 2$ be a natural number. A subset $A$ of $\mathbb{R}$ is called $n$-small if there exist $n$ real numbers $t_1, t_2, \ldots, t_n$ such that the sets $t_1 + A, t_2 + A, \ldots, t_n + A$ are pairwise disjoint. Prove that $\mathbb{R}$ cannot be represented as a union of $n$ $n$-small sets.
Notation: If $r \in \mathbb{R}$ and $B$ is a subset of $\mathbb{R}$, then $r + B = \{ r + b \mid b \in B \}$
Notation: If $r \in \mathbb{R}$ and $B$ is a subset of $\mathbb{R}$, then $r + B = \{ r + b \mid b \in B \}$