MathNet 2020 — Math Olympiad Vault — Part 03 Question 7

Let $n \gt 1$ be a positive integer and $S$ be a collection of $\frac{1}{2}\binom{2n}{n}$ distinct $n$-element subsets of $\{1,2, \ldots, 2n\}$. Show that there exists $A, B \in S$ such that $|A \cap B| \leq 1$.