Let $S$ be a set of size $1983$, and let $A_1,..,A_k$ be a familiy of subsets of $S$ such that:
- The union of every 3 sets of the family is S.
- For every pair of sets the union of them contains no more than $1979$ elements of $S$.
What is the maximal size of $k$?
I'm pretty stuck here, can someone give me a hint on how to solve it?
Hint: Let $B_i$ be the complement of $A_i$. Now each pair $B_i, B_j$ have intersection $C_{ij}$ with at least $4$ elements, and $C_{ij} \cap C_{kl}=\emptyset$ for $\{i, j\} \not =\{k, l\}$.