Let $S$ be a set of some size $n$. I'm interested in knowing about combinatorial designs that are approximately balanced incomplete block designs, that I want a collection of subsets $C$ of $S$ such that every pair of elements of $S$ appears in at least one element of $C$, all the elements of $C$ are approximately the same size, and every element appears approximately the same number of times. Obviously there are many trivial examples, but those aren't interesting. Ideally given $n$ there would be a set where $C$ is not "too large" and the elements of $C$ are also not "too large", for some definition of "too large". Most of the literature I've looked at considers weaker exact conditions.

I'm asking this question because I have a massive computation to carry out on multiple machines, and the computation considers a pair of big files at a time. I want to minimize the number of files each of my workers read, while also keeping down the number of workers. Papers, proofs, guidance on what I should be searching for are all welcome: my combinatorial knowledge is shamefully weak.