How to use the intersection triangle of the Steiner System $S(5,6,12)$ to show the completement of any of its block is also a block?
My argument: Since the parameter $m_{60}$ in the intersection triangle is 1, so each block $B$ has a unique block $B'$ which is disjoint to the former, thus $B' \subseteq X\diagdown B$. Moreover, since $|X\diagdown B|=12-6=6=|B'|$, so $X\diagdown B=B'$ which implies $X\diagdown B$ is a block. Is my argument valid based on my interpretation of $m_{60}$?
For the convention of the subscripts please Google the book “Topics on Steiner Systems” By Charles C. Lindner, A. Rosa, on page 49