Allocating balls among boxes under specified conditions

Question

There are $k$ boxes to which a subset of $nk$ balls are to be allocated. The full set of $nk$ balls consists of $k$ balls numbered $1$, $k$ balls numbered $2$, ... , $k$ balls numbered $n$. Each box must be allocated exactly $m$ different balls (i.e. no two balls in a given box can have the same number). What conditions must be imposed on $n$, $k$, and $m$ so that any pair of boxes have at least one ball with the same number (i.e. for any two boxes chosen, there is some number $j$, $1 \leq j \leq n$, such that ball $j$ exists in each of the two boxes)?

Answer 1

It sounds to me like you need $m \gt \frac n2$. Then the pigeonhole principle tells you that there must be a matching number between any pair of boxes. If $m=\frac n2$ box $1$ could have all the odd numbers and box $2$ could have all the even numbers so you would fail, showing the bound is tight.