Let $A=\left\{ {1, 2, \ldots, n}\right\}$. Let $B$ be the set of all size $m$ subsets of $A$. $B=\left\{{B_1,B_2, \ldots , B_{\binom{n}{m}} } \right\}$, $ |B_i|=m$ then we want to find $k$ subsets of $A$, each of size $r$, (let call them $C=\left\{{C_1,C_2,\ldots,C_k}\right\}$, $|C_i|=r$ ), so that for any $B_i$, there exists at least one $C_j$ in $C$ so that the intersection of $B_i$ and $C_j$ is empty.
Example, if $n=7$, $m=3$, $r=2$, then with $k=7$ subsets we can build $C$ as follows: $C=\left\{ {\left\{ {1, 2}\right\}, \left\{ {2, 3}\right\}, \left\{ {3, 4}\right\}, \left\{ {4, 5}\right\}, \left\{ {5, 6}\right\}, \left\{ {6, 7}\right\}, \left\{ {7, 1}\right\}}\right\}$
which has the property that for any subset of $A$ of size $m=3$, denoted by $B_i=\left\{{b1,b2,b3}\right\}$, there is at least one $C_j$ in $C$, that has no intersection with $B_i$.
The question is given $n$,$m$, and $r$, what is the minimum $k$ for which a corresponding $C$ exists?
If you replace the sets $C_j$ by their complements $C_j'$, then each $B_i$ must be contained in some $C_j'$. This is called a covering design. The La Jolla Covering Repository is a great place to look.