SMO 2006 Problem 4

Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$ -element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$ . Show that $$k\leq 6^{(n+1)/2}$$