Assume we have a collection of $2^k$ different items and we need to pick $2k$ sets of different items out of it under the following constraints:
- Each item must be picked at least in $2$ sets.
- The intersection of at least $2$ different sets in which an item I appears contains only I itself, Si ∩ Sj = I. The complementary sets of the picked sets should also obey this.
My approach is to line up the items in $k$ rows and run an initial pick of k sets with $2^k$ in each (the last one may be partial). Next i tried to formulate picking the other k sets by selecting extended diagonals from the above setting. While i believe this approach may be correct, i could not prove it for any value of $k$.
Help appreciated, thank you in advance.
Imagine your sets as nodes $n_i=S_i$ of a complete graph and the intersections between sets as edges $e_{ij}=S_i\cap S_j$. Since you have $2k$ nodes, you will also have $k(2k-1)$ edges.
For each of the $2^k$ elements, you want to have a dedicated edge $e_{ij}=I$. If $k=1$ or $k>6$, $k(2k-1)<2^k$. Thus, it's not possible in general.