I would like to partition a set ${1, 2, ..., n}$ into $k$ equally sized subsets, and perform this operation $b$ times. In the end, my aim is to end up with a situation where for each given pairwise combination of set members, e.g. {1, 2}, the numbers occurs together in a subset in no more than $x$ of the $b$ partitions.
For example, suppose I have a set of 100 numbers, I want to find out what how many subsets ($k$) I need to achieve the following goal if I partition the set $b$ times:
| Partition | membership of pair ${1,2}$ | ... | membership of pair ${99, 100}$ |
|-----------|--------------------------------|-----|-----------------------------------|
|1 | *${1, 2} \in$ any subset $k$* | ... | ${99, 100} \notin$ any subset $k$ |
|2 | *${1, 2} \in$ any subset $k$* | ... | *${99, 100} \in$ any subset $k$* |
|... | ${1, 2} \notin$ any subset $k$ | ... | *${99, 100} \in$ any subset $k$* |
|b | ${1, 2} \notin$ any subset $k$ | ... | ${99, 100} \notin$ any subset $k$ |
In other words, every pair of numbers needs to occur together in a single subset of the $b$ partitionings at most $x = 2$ times, and be disjoint in all the other partitions. So for a given pair ${1,2}$, the numbers will only occur together in a subset in partition 1 and 2. In all other partitions the numbers will be in separate subsets. The same goes for every other pair of numbers (columns). It does not matter in which specific iteration each pair is joint or disjoint, as long as there exist at most two iterations where they are together.
Ideally, the total number of iterations $b$ would be fixed, and I want to find the number of subsets $k$ required to ensure the "each pair occurs in a subset no more than $x$ times across partitions"-property is fulfilled.
Is this a known problem? I attempted to search for related set theory problems, but I quickly got lost in the terminology since I am not familiar with it. The closest related problem seems to be something along the lines of "a family of almost-disjoint families of sets", where the inner families are partitions and the almost-disjoint property applies to pairs occurring together or not within each inner family, although I might be abusing existing terminology here.
Suppose $n/k=m$, so everysubset has $m$ elements and each pair joint exactly $x$ times. Now we count the number of each elements in two other way:
First It appears $b$ times obviously.
Second It appears $x$ times with every other element. Every element in each subset pair with $m-1$ other elements, and so count $m-1$ times, therefore $x\binom{n}{2}/(m-1)$
So we have: $$xn(n-1) = b(m-1)$$ Now if you change "Exact $x$ Times" to "At Most $x$ Times", the quality become inequality.
For more details see:
https://en.wikipedia.org/wiki/Block_design