Techniques for showing that the common intersection of a family of sets is non-empty Suppose that I have a family of sets $\mathcal{F}$ from a universe of size $n$, where each set in $\mathcal{F}$ has size $r$. I am interested in results that allow me to obtain a lower bound on the size of the intersection of every set in $\mathcal{F}$. Now I know that in most cases, that this intersection will be empty, but I would like to know theorems or results that state required conditions that guaranty that this intersection will be non-empty (or a non-trivial lower bound). In other words, what do we need to know about $\mathcal{F}$ such that this holds? I am asking this with regards to an earlier question I posted, namely Bounding the global intersection of a family of sets - Hopefully this shorter question may be easier to answer.

If $\mathscr F$ has $m$ members and $m(n-r) \ge n$, then it is possible for the intersection to be empty, since  the universe can be written as the union of $m$ sets of size $n-r$, and you can take $\mathscr F$ to be
the  complements of these.
