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.



Constructive ideas behind "covering" a set with subsets of fixed size
You are looking for [covering designs](https://www.ccrwest.org/cover.html). In general, it is very hard to determine these numbers exactly. A lot of work has been done on bounding them ([Gordon, Kuperberg, Patashnik](http://arxiv.org/pdf/math/9502238): "Hundreds of papers have been written for particular values of $v$, $k$, and $t$."). Some references for the case $t=2$ are


* [Todorov (1989)](http://dx.doi.org/10.1007/BF02124682)
* [Furedi (1990)](http://dx.doi.org/10.1016/0097-3165%2890%2990034-T)
* [Bryant, Buchanan, Horsley, Maenhaut, Scharaschkin (2011)](http://dx.doi.org/10.1007/s00493-011-2639-y)


