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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.

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A small collection of large subsets covering all small subsets.

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.