This question came up in a practical situation at work...


Given a set S of size n, what is the minimum number of subsets of S of size k s.t. each pair of elements of S occurs in the same subset at least once?



minimum number of subsets?
If you want each $l$-tuple of elements to occur at least once then you can do this with $(1 + o(1))\binom{n}{l}/\binom{k}{l}$ sets, which is within $1 + o(1)$ of optimal by a simple counting argument. This is proven using something called the Rödl Nibble. Most likely when $l = 2$, which is the situation you're interested in, there is a direct construction. The problem is perhaps more naturally formulated in terms of graphs: you're looking for a covering of all the edges of the complete graph $K\_n$ by copies of $K\_k$.


In terms of explicit constructions, this paper seems relevant. I'm led to understand that one of the authors takes a passing interest in Math Overflow; perhaps he will comment further.


* [MR1333298](https://mathscinet.ams.org/mathscinet-getitem?mr=1333298)
Daniel M. Gordon, Oren Patashnik, Greg Kuperberg,
*New constructions for covering designs.*
J. Combin. Des. **3** (1995), no. 4, 269–284, doi:[10.1002/jcd.3180030404](https://doi.org/10.1002/jcd.3180030404).


In the spirit of current discussions on open access, a free version of the above is at <https://arxiv.org/abs/math/9502238>


