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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?

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Yes, this is an active research area. For a quick survey, see the introduction to this paper of Furedi and Sudakov. (they also prove many interesting extensions).

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