let $A=\{1,2,3...,N\}$ and $B\_1,B\_2,B\_3\dots,B\_n$ be a series of subsets of $A$ which satisfied that $|B\_i|=m$, 
$|B\_i\cap B\_j|\le k$. what is the maximum of $n$? ($k< m< N$)


it can be easily showed that 
$n\le [C(N−1,k)/C(m−1,k)]/[N/m]$ (by counting twice, $[X]$ is integer part of $x$)


I wonder is there any reserch that tackle this problem? Do we have some profound result?



set and subset series combinatorics
Yes, this is an active research area. For a quick survey, see the introduction to [this paper of Furedi and Sudakov.](http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_sudakov_kwise.pdf) (they also prove many interesting extensions).


