Question 7
Let $n \geq 5$ be an integer, and $E_{1}, E_{2}, \ldots, E_{2 n-1}$ be distinct two-element subsets of $\{1,2, \ldots, n\}$.
Prove that, among the $2 n-1$ subsets $E_{i}$, one can choose $n$ of them so that the union of these $n$ subsets contains at most $\frac{2}{3} n+1$ elements.
Prove that, among the $2 n-1$ subsets $E_{i}$, one can choose $n$ of them so that the union of these $n$ subsets contains at most $\frac{2}{3} n+1$ elements.