Question 62
Let $n, s, t$ be three positive integers, and let $A_1, \dots, A_s, B_1, \dots, B_t$ be subsets of $\{1, 2, \dots, n\}$ (not necessarily distinct). For any subset $S$ of $\{1, \dots, n\}$, define $f(S)$ as the number of $i \in \{1, \dots, s\}$ with $S \subseteq A_i$, and $g(S)$ as the number of $j \in \{1, \dots, t\}$ with $S \subseteq B_j$. Suppose that $f(\{x, y\}) = g(\{x, y\})$ for all $1 \le x \lt y \le n$.
Prove that if $t \lt n$, then there exists $1 \le x \le n$ such that $f(\{x\}) \ge g(\{x\})$.
Prove that if $t \lt n$, then there exists $1 \le x \le n$ such that $f(\{x\}) \ge g(\{x\})$.