Question 60
Let $n$ be an integer greater than $1$. The set $S$ of all diagonals of a $(4n-1)$-gon is partitioned into $k$ sets, $S_1, \dots, S_k$, so that, for every pair of distinct indices $i$ and $j$, some diagonal in $S_i$ crosses some diagonal in $S_j$; that is, the two diagonals share an interior point. Determine the largest possible value of $k$ in terms of $n$.
