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If $n=\frac{p^t-1}{p-1}$ and $k=p+1$ then we can consider the set $N$ as a finite projective space and our sets be a lines in this space.
I think that in other cases problem is hard.
If $n=\frac{p^t-1}{p-1}$ and $k=p+1$ then we can consider the set $N$ as a finite projective space and our sets be lines in this space.
