Question 19
Determine the greatest possible value of $n$ that satisfies the following condition: for any choice of $n$ subsets $M_1, ..., M_n$ of the set $M = \{1, 2, ..., n\}$ satisfying the conditions
i) $i \in M_i$; and
ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \neq j$,
there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$.
i) $i \in M_i$; and
ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \neq j$,
there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$.