Question 58
Let $E$ be a given set with $n$ elements. Suppose that $A_1, A_2, \dots, A_k$ are $k$ distinct nonempty subsets of $E$, with either $A_i \cap A_j = \emptyset$, $A_i \subset A_j$, or $A_j \subset A_i$ for any $1 \le i \lt j \le k$. Find the maximum value of $k$.