Question 181
Let $m$ be a positive integer and $A$ be a finite set. Let $A_1, A_2, \dots, A_m$ be subsets of $A$ (not necessarily distinct). It is known that for any nonempty set $I \subseteq \{1, 2, \dots, m\}$,
$$
\left| \bigcup_{i \in I} A_i \right| \geq |I| + 1.
$$
Prove: the elements of $A$ can be coloured black or white, such that every of $A_1, A_2, \dots, A_m$ contains both black and white elements.
$$
\left| \bigcup_{i \in I} A_i \right| \geq |I| + 1.
$$
Prove: the elements of $A$ can be coloured black or white, such that every of $A_1, A_2, \dots, A_m$ contains both black and white elements.