2008 IMO Shortlist 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.

对于 $n \geq 2$，设 $S_{1}、S_{2}、\ldots、S_{2^{n}}$ 为 $A=\left\{1,2,3, \ldots, 2^{n+1}\right\}$ 的 $2^{n}$ 子集，且满足以下性质：不存在索引 $a$ 和 $b$ 且 $a<b$ 和元素 $x, y, z \in A$ 且 $x<y<z$ 使得 $y, z \in S_{a}$ 和 $x, z \in S_{b}$。证明集合 $S_{1}, S_{2}, \ldots, S_{2^{n}}$ 中至少有一个包含不超过 $4 n$ 个元素。