Consider all the $\left(\begin{array}{c} n\\k\end{array}\right)$ possible combinations of $k$ elements from a fixed set of $n$ elements. Let’s now form groups of these combinations with the following property: In each group there exist at least one combination with at least $s<k$ elements in common with all other combinations of the group. Is it possible to find the minimum number of groups that cover all the $\left(\begin{array}{c} n\\k\end{array}\right)$ combinations, as a function of $n$, $k$ and $s$? The exact number would be great, but some (sharp) upper limit would also be very useful.
In other words, I would like to find the $s$-net covering number of the set of all $\left(\begin{array}{c} n\\k\end{array}\right)$ possible combinations, with respect to the distance metric corresponding to the number of non-overlapping elements.