How about $n=7$, so $N={3\choose 2} = 3$, with subsets $S_1 = \{ 1,2,3,4\}$ and $S_2 = \{ 1,2,5,6\} $ and $S_3 = \{ 3,4,5,6 \} $ and $S_4 = \{ 1, 3, 5, 7 \} $.
Patricia Hersh
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