RO-TST 2013 Problem 4

Let $n$ be an integer greater than 1. The set $S$ of all diagonals of a $\left( 4n-1\right)$ -gon is partitioned into $k$ sets, $S_{1},S_{2},\ldots ,S_{k},$ 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 $n.$