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The question is the following: how many subsets of size $5$ from a set $A$ of size $16$ do we need so that any subset of size 2 of $A$ is also a subset of one of the selected subsets of size $5$?

How does this the required number change as we change 16 to another number and if we change $5$ to another number? Perhaps an even harder question is if we change $2$ to another number.

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You are looking for covering designs. In general, it is very hard to determine these numbers exactly. A lot of work has been done on bounding them (Gordon, Kuperberg, Patashnik: "Hundreds of papers have been written for particular values of $v$, $k$, and $t$."). Some references for the case $t=2$ are

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  • $\begingroup$ Is there more knowledge about coverings where t=2? I'm very interested in that case.(where t is the size of the subsets that need to be covered) $\endgroup$
    – Gorka
    May 20, 2014 at 14:56

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