Are there any known results about the following problem:

Given any integer $n\geqslant4$, how many $4$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any two $4$-elements subsets we have choosed has not more than $2$ element?

More generally, given any integer $n\geqslant2d$, how many $2d$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any two $4$-elements subsets we have choosed has not more than $d$ element?