Let $r,s,n$ be positive integers with $r < s < n$. Let $U = \{1,\ldots,n\}$.



> 
> Let $S$ contain $s$-element subsets of
>  $U$ (of our choosing). What is that smallest we can make $S$ such that every
>  $r$-element subset of $U$ is a subset
>  of some element in $S$?
> 
> 
> 


I'm curious what the best known upper and lower bounds are on the smallest we can make $S$, and I am especially interested in the case where $r << s << n$. I am more worried about asymptotic behavior than exact bounds.


This can also be viewed as a special case of a set-cover problem that has a lot of symmetry. I'm afraid/hoping there's something simple from graph theory that solves my problem.


EDIT: Thanks to @Gerhard, I see this is a well known problem called covering numbers / covering designs.



A small collection of large subsets covering all small subsets.
What you are looking for is an $(r,s,n)$-covering design. A good starting point might be the [La Jolla Covering Repository](http://www.ccrwest.org/cover.html) or in fact, any book on design theory. In general, the smallest possible size for your covering design is given by $\binom{n}{r}/\binom{s}{r}$ in which case every $r$ -element subset is contained in exactly one $s$-element subset and the design is then called a $(r,s,n)$-Steiner system. There are necessary conditions for Steiner systems to exist such as $\binom{s-i}{r-i} | \binom{n-i}{r-i}$ for all $i$. Quite a lot is known about the existence of $(2,3,n)$-Steiner systems, and $(3,4,n)$-Steiner systems, but in general their existence is an open question.


