# Assignment of subsets with overlap rule

I want to create $$M$$ lists of size-$$k$$ subsets of $$\mathbb{Z}_n$$, labeled $$A_1, \ldots, A_M$$ with the following properties:

1. $$|A_1| = |A_2| = \ldots = |A_M|$$, i.e. every list contains the same number of subsets.
2. $$\forall (a_1 \in A_1, \ldots, a_M \in A_M) \left|\bigcap_{i=1}^M a_i \right | = 1$$, i.e. if you pick one subset from every list, those subsets will have exactly one element in common.
3. I want the size of the lists to be as large as possible.
4. I want, in some sense, the distribution of the common elements to be as even as possible.

To give an example, let $$n = 4$$ and $$M = k = 2$$. Then I could create $$a_1 = \left\{ \{0, 1\}, \{2, 3\} \right\}$$ and $$a_2 = \left\{ \{0, 2\}, \{1, 3\} \right\}$$, since the intersection of any set in $$a_1$$ and any set in $$a_2$$ gives a different singleton.

By comparison, if $$n = 6$$, $$k = 3$$ and $$M = 2$$, then I could create $$a_1 = \left\{ \{0, 1, 2\}, \{0, 1, 5\} \right\}$$ and $$a_2 = \left\{ \{0, 3, 4\}, \{1, 3, 4\} \right\}$$ and that still satisfies 1 and 2, but for 3 I don't know whether I could actually find a solution where each list has 3 elements, and for 4 the values of the intersections are $$0, 0, 1, 1$$ which is very skewed since there are 2 repeats and 3 values not appearing. I can improve the latter by making the lists $$a_1 = \left\{ \{0, 1, 2\}, \{1, 3, 5\} \right\}$$ and $$a_2 = \left\{ \{0, 3, 4\}, \{2, 4, 5\} \right\}$$ since then the intersections are $$0, 2, 3, 5$$.

Is there a method or algorithm that would give me a set of lists that satisfy 1 and 2, along with some level of confidence that I've done as well as I can on points 3 and 4? Are there any special cases where I can get provably optimal results easily?

I note that this is similar to the question about creating Dobble cards, but in that case the aim is for every pairwise intersection to be a singleton whereas I'm looking to create these lists of subsets.

• Question: In (3), does it mean $M, k, n$ are all given and you want to maximize $|A_i|$? Or can you, during the maximization, vary some of $M, k, n$? Comment: the $M=2$ case reminds me a lot of finite planes, where the subsets are lines and the elements are points. Then (2) becomes: two lines picked one from each list must intersect (at one point), so if you're basing on finite projective planes this follows immediately, whereas if you're basing on finite affine planes then this also follows if $A_1$ and $A_2$ do not contain two lines which are parallel. – antkam Jun 13 at 18:02
• Look up finite (projective / affine) planes in wikipedia. Such planes are known to exist for certain values of $k, n$ and they would provide examples for the $M=2$ case. Many such adopted examples should do very well re: (4), but it's less clear to me if they optimize (3). – antkam Jun 13 at 18:04
• Bloody affine planes, whenever I come up with something like this they seem to be the answer and I probably need to properly learn about them. As for the question - the main aim is to optimise the size of the lists, but then within that I want to get the even spread of cross-over. – ConMan Jun 18 at 23:53