Question 14
Let $A_1, A_2, A_3, \dots, A_n$ be $n$ subsets of a finite set $S$ such that $|A_j| = 8$ for each $j$, $1 \le j \le n$. For a subset $B$ of $S$, let $F(B) = \{j : 1 \le j \le n \text{ and } A_j \subset B\}$. Suppose for each subset $B$ of $S$, at least one of the following conditions holds:
(i) $|B| \gt 25$;
(ii) $F(B) = \emptyset$;
(iii) $\cap_{j \in F(B)} A_j \ne \emptyset$.
Prove that $A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n \ne \emptyset$.
(i) $|B| \gt 25$;
(ii) $F(B) = \emptyset$;
(iii) $\cap_{j \in F(B)} A_j \ne \emptyset$.
Prove that $A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n \ne \emptyset$.