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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?

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This can be thought of as finding the largest binary code of constant weight $2d$, length $n$, minimal distant $2d$.

For $d=2$ the values are A001843 in the OEIS. References/links on the OEIS page have some further information about both $d=2$ and other values of $d$.

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    $\begingroup$ John Machacek: Thanks for your answer! $\endgroup$
    – user173856
    Sep 26, 2018 at 16:33

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