I have a ground set $A$ of $n$ elements. I want to create a collection $C$ of $m$ sets of size $k < n$ such that every subset $S\subseteq A$ of size $k$ is covered by at most $t$ sets from $C$ (i.e., every set $S$ is contained in a union of at most $t$ sets from $C$). What is the minimum value of $m$ such that such a collection exists as a function of $t$ (and $n,k$)? For example, for $t=1$ we need all such subsets so $m= {{n} \choose{k}} $, and for $t=n/k$ we can create an arbitrary partition so we need $m=\lceil n/k\rceil$.
In particular, what is $m$ for $t \in \{2,\log(n/k),\sqrt{n/k}\}$?
