SMO 2006 Open Round 2 Question 4

Let $n$ be a positive integer. Let $S_1,S_2,\ldots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,\ldots,4n-1,4n\}$ such that $S_i\cap S_j$ contains at most $n$ elements for all $1\le i\lt j\le k$. Show that
$$k\le6^{(n+1)/2}.$$