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 a lines in this space.
I think that in other cases problem is hard.