Question 55
In an art museum, $n$ paintings are exhibited, where $n \ge 33$ is a positive integer, and a total of 15 colors are used in such a way that any two paintings have at least one color in common, and no two paintings have exactly the same set of colors. Determine all possible values of $n \ge 33$ such that, no matter how we color the paintings with the above properties, we can choose four distinct paintings that we number $T_1, T_2, T_3$, and $T_4$ in such a way that any color used both in $T_1$ and $T_2$ is also used in $T_3$ or $T_4$.