The goal is to obtain the minimum number of subsets of size s from a set of items of size a, so that pair of items should appear in exactly n subsets

Question

I am not a mathematician, I hope the question is sufficiently clear and has a solution.

The use case is the design of an online experiment.

• Parameter a is typically between 10 and 1000.
• Parameter s is typically between 3 and 6.
• Parameter n is typically between 1 and 20.

If a formal solution does not exist or is very computationally intensive, an algorithmic solution that would give an approximate, faster answer would be great as well (list of subsets where almost all pairs of items appear in n subsets).