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Let $S_i$ be a subset of $S=\{1,2,\ldots,n\}$ for $i = 1,2,\ldots,n-1$. Prove that there exists a nonempty subset $R$ of $S$ such that $|S_i\cap R|\neq1$ for each $i=1,2,\ldots,n-1$.

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    Why does taking $R = S$ not work? (Perhaps I don't understand the meaning of $(n-1)$ subset, but I think it is just a subset with $n-1$ elements.)2012-03-09
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    Also, regardless of what the $S_i$ have to satisfy, $R = \emptyset$ trivially works.2012-03-09
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    sorry! I did not make this problem clear. First, n-1 means there are n-1 subset $S_1,\cdots,S_{n-1}$, and it does not mean $S_i$ has n-1 elements. Second, we assume $R$ is not null set. I am sorry for that.2012-03-09

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