Question 86
Let $n \ge 2$ be an integer and $E = \{1, 2, \dots, n\}$. If $A_1, A_2, \dots, A_k$ are subsets of $E$ and exactly one of $A_i \cap A_j$, $A'_i \cap A_j$, $A_i \cap A'_j$ and $A'_i \cap A'_j$ is empty for all $1 \le i \lt j \le k$, then determine the maximum possible value of $k$.
[For $A \subset E$, $A'$ denotes the elements of $E$ which are not included in $A$.]
[For $A \subset E$, $A'$ denotes the elements of $E$ which are not included in $A$.]