KöMaL 2021 A811

Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\dots,m$ satisfying the following conditions: $(i)$ $B_i \subset A$, $|B_i|=k$, $(ii)$ $C_i \subset B_i$ (there is no additional condition for the number of elements in $C_i$), $(iii)$ $B_i \cap C_j \ne B_j \cap C_i$ for all $i \ne j$.