KöMaL 2025 C1875

Thanks to a generous donation, during the lunch break of the KöMaL Ankét on Saturday, $m$ types of books were distributed among the $n$ students present. There were multiple copies available of each book type, and everyone could choose freely: they could take books of several types, but at most one copy of each type. We know that for any two books there were a difference of at least three students between the set of students choosing each book. Prove that $m \leq \frac{2^{n}}{n+1}$.