Question 167
Determine the largest natural number $n$ for which there exist distinct sets $S_{1}, S_{2}, \ldots, S_{n}$ such that:
$1^{\circ}\left|S_{i} \cup S_{j}\right| \leqslant 2004$ for any two integers $1 \leqslant i, j \leqslant n$, and
$2^{\circ} S_{i} \cup S_{j} \cup S_{k}=\{1,2, \ldots, 2008\}$ for any three integers $1 \leqslant i\lt j\lt k \leqslant n$.
$1^{\circ}\left|S_{i} \cup S_{j}\right| \leqslant 2004$ for any two integers $1 \leqslant i, j \leqslant n$, and
$2^{\circ} S_{i} \cup S_{j} \cup S_{k}=\{1,2, \ldots, 2008\}$ for any three integers $1 \leqslant i\lt j\lt k \leqslant n$.