Assume I have a universe of N elements.
The question is: How many sets of size $X$ are needed to assure that every set of K elements is a subset of (at least) one of these sets (where $K \ll X \lt N$). And also, how can these sets be chosen to obtain this minimum?
In particular, the sizes that interest me are: \begin{align*} N &= O(2^{n^{c_1}}),\\ X &= O(2^n),\\ K &= O(n^{c_2}), \end{align*} where $n$ is a variable, $c_1$ and $c_2$ are constants.