We are given universe ${1,...,n}$ and all its $r$-sized subsets. I want to find a family of bijections $\phi_{A,B}:A\mapsto B$ between every two $r$-sized subsets $A,B$, with the following property:
0) mappings are symmetric, i.e., $\phi_{B,A}$ is just reversed $\phi_{A,B}$,
1) for any two subsets $A,B$, if $e \in A \cap B$, then $\phi_{A,B}(e) = e$,
2) for any three subsets $A,B,C$ and $a\in A, b\in B, c\in C$, if $\phi_{C,A}(c) = a$ and $\phi_{C,B}(c) = b$, then $\phi_{A,B}(a) = b$.
It would be really easy to construct such a family if we would waive property 1 --- then just order all subsets in some arbitrary way, and map to each other all elements of the same number in the subset. Question is can we still have this transitivity property 2, if we insist on 1.
