ISL 2008 C6

For $n \geq 2$, let $S_{1}, S_{2}, \ldots, S_{2^{n}}$ be $2^{n}$ subsets of $A=\left\{1,2,3, \ldots, 2^{n+1}\right\}$ that satisfy the following property: There do not exist indices $a$ and $b$ with $a<b$ and elements $x, y, z \in A$ with $x<y<z$ such that $y, z \in S_{a}$ and $x, z \in S_{b}$. Prove that at least one of the sets $S_{1}, S_{2}, \ldots, S_{2^{n}}$ contains no more than $4 n$ elements.