My colleague is investigating the following problem.
For a given natural number $n$ construct a specific balanced block design, namely, a family $\mathcal D$ consisting of $n$-element subsets of a set $X$ of size $2n$ such that each element of $X$ is contained in exactly a half of members of the family $\mathcal D$ and for each $n$-element subset $S$ of $X$ exactly one of sets $S$ and $X\setminus S$ belongs to $\mathcal D$.
For instance, for $n=3$ we can pick as $\mathcal D$ the following family of subsets of a set $\{1,\dots, 6\}$.
$$124\; 235\; 346\; 451\; 562\; 613\; 123\; 345\; 561\; 246$$
Let $[X]^n$ be the family of all subsets of $X$ of size $n$. Since $|\mathcal D|=|[X]^n |/4$, the required family exists only provided $|[X]^n |={2n\choose n}$ is divisible by $4$. This holds iff $n$ is not a power of two.
Thus the problem is how to construct a required design for other values of $n$. We are amateurs in this subject, so my solving experience of such problems suggests the following approach.
First try to solve the problem for small values of $n$. He constructed the required design for $n=3, 5, 7, 9$, grouping the members of $[X]^n$ into symmetry classes and orbits and then picking to $\mathcal D$ some of them, balancing by hand (the detailed description is too long to be presented here). The first hard case turned out to be $n=6$, for which his approach failed. Nevertheless, I didn’t find a parity or divisibility conflict which often assures impossibility of a combinatorial construction. Thus his first question is: whether there exists a required block design for $n=6$?
For the next step solving such problem we need a luck, because we try to find a general pattern providing the required construction. A typical examples are finite planes providing Steiner triple systems. If we are not so lucky we still can try construct a solution balancing several different patterns. Unfortunately, already for $n=6$ the family $|[X]^n|$ consists of $924$ sets and today is not my lucky day.
I can write a program, starting from a random subfamily $\mathcal D$ of $|[X]^n|$, such that for each $S\in [X]^n$ exactly one of sets $S$ and $X\setminus S$ belongs to $\mathcal D$ and then iteratively trying to balance the family $\mathcal D$ assuring that each element of $X$ is contained in exactly a half of members of $\mathcal D$. But I guess I shall not be able to generalize the found solution to other values of $n$, because I expect it will be too random to have a pattern.
Finally, since my colleague hopes to publish his research, and for me it is hard to formulate a relevant Google query, his second question is about references for the problem. Was it studied before, are there known results? I remark that the problem can be related to hypergraphs, because a required design provides a regular $n$-uniform hypergraph.
Thanks.