Is there a method for determining whether a Steiner system contains a subset of smaller Steiner systems?
For example, consider the Steiner Triple System $S(2, 3, 9)$. A valid output of this could be the following:
[1, 2, 3]
[1, 4, 8]
[1, 5, 7]
[1, 6, 9]
[2, 4, 7]
[2, 5, 6]
[2, 8, 9]
[3, 4, 6]
[3, 5, 8]
[3, 7, 9]
[4, 5, 9]
[6, 7, 8]
Within $S(2, 3, 9)$ there also exists $4$ systems of $S(1, 3, 9)$:
[[1, 4, 8], [2, 5, 6], [3, 7, 9]]
[[1, 2, 3], [4, 5, 9], [6, 7, 8]]
[[1, 6, 9], [2, 4, 7], [3, 5, 8]]
[[1, 5, 7], [2, 8, 9], [3, 4, 6]]
Assuming that the number of blocks, $b$, is evenly divisible by $n/k$, it seems logical to me that all Steiner systems of $S(t, k, n)$ should, at the very least, contain $S(t-1, k, n)$ within them. This seems especially logical in the case of when $t = 2$.
For example:
- $S(2, 3, 9)$
- $b = 12$
- $k/n = 9/3 = 3$
- $b/(k/n) = 12/3 = 4$
I am having trouble determining a way to prove this is always the case, and whether it is true for the cases that I am investigating.
The Steiner system that I am investigating is $S(2, 6, 96)$. At a basic level it seems that that the $304$ blocks for $S(2, 6, 96)$ should be able to be divided into $16$ systems of $S(1, 6, 96)$.
I believe another term for this is that the Steiner system is resolvable. There appears to be literature on whether an STS is resolvable, but I cannot seem to find any information on whether other Steiner systems are resolvable. Outside of creating a computer search, is there a method or formula for determining whether a solution to this problem exists?