Question 91
Let $n$ be a positive integer, and let $S_1, \dots, S_n$ be a collection of finite non-empty sets such that
$$
\sum_{1 \le i \lt j \le n} \frac{|S_i \cap S_j|}{|S_i| |S_j|} \lt 1.
$$
Prove that there exist pairwise distinct elements $x_1, \dots, x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.
$$
\sum_{1 \le i \lt j \le n} \frac{|S_i \cap S_j|}{|S_i| |S_j|} \lt 1.
$$
Prove that there exist pairwise distinct elements $x_1, \dots, x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.