Is there a nice way to partition the edges of the complete $5$-uniform hypergraph
on $11$ vertices into $7$ copies of the Steiner system $S(4,5,11)$? If this is
obvious or elementary, I apologize in advance.



On the Steiner system $S(4,5,11)$
Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ disjoint copies of an $S(4,5,11)$ in the complete $5$-uniform hypergraph on $11$ vertices (or partition it into copies); you can find only two of them at most.


* E. S. Kramer, D. M. Mesner, *Intersections among Steiner systems*, *J. Combin. Theory, Ser. A*, **16** (1974), 273–285, <https://doi.org/10.1016/0097-3165(74)90054-5>


