Question 94
Let $A_1, A_2, \dots, A_n$ be sets. For a subset $X$ of $\{1, 2, \dots, n\}$, let
$$
N(X) = \{i \in \{1, 2, \dots, n\} - X : A_i \cap A_j \neq \emptyset \text{ for all } j \in X\}.
$$
Prove that for every integer $3 \le m \le n - 2$, there exists a subset $X$ of $\{1, 2, \dots, n\}$ such that $|X| = m$ and $|N(X)| \neq 1$.
$$
N(X) = \{i \in \{1, 2, \dots, n\} - X : A_i \cap A_j \neq \emptyset \text{ for all } j \in X\}.
$$
Prove that for every integer $3 \le m \le n - 2$, there exists a subset $X$ of $\{1, 2, \dots, n\}$ such that $|X| = m$ and $|N(X)| \neq 1$.