Given integers $ n,r,s $, we can define $M(n,r,s)$ to be the maximal size of a family $F$ of $r$-subsets of $\{1,...,n\}$ such that the pairwise intersection between any two subsets is at most $s$. Is there any way to get a (non-trivial) upper bound on $M(n,r,s)$?


If we want the intersections to be at least $s$ (instead of at most $s$), we have a generalization of the Erdos-Ko-Rado problem, and this question has been asked before on this site. 


I am chiefly interested in the case where $r$ is linear in $n$ and $s=\frac{r}{2}$.



An upper bound on families of subsets with a small pairwise intersection
The maximum size is attained by a Steiner system $S(t+1,r,n)$ when it exists. It consists of $\binom{n}{t+1}/\binom{r}{t+1}$ blocks.


See <http://en.wikipedia.org/wiki/Steiner_system>


