Let $N = \{1, 2,\ldots, n\}$. Let $\{A_1,A_2, \ldots, A_m\}$ be subsets of N and there union is N. What is the minimum value of m for which for any $Y\subseteq N$ ($Y\neq A_i$ for any i), there exists $A_i$ for some $i\in \{1,2, \ldots,m\}$ such that $|Y\cap A_i|=1$?
I found that $\{1,2\},\{3\}, \{4\}, \ldots, \{n\} $ is such a collection. So m=n-1 here. Is this the minimum?