Does there exist an $(11,3,3)-BIBD$ such that:
No element of $B$ is repeated where $B$ is the set of blocks.
There exists a subset $B'$ of $B$ such that any pair of elements of $\{1,2,...,11\}$ is in exactly 2 blocks of $B'$.
My thoughts on this are no. But it seems too easy. We know that in order for this design to exist there needs to be an $(11,3,2)-BIBD$ since $B'$ is just a subset of $B$. But we know by the well-known theorems that $r=10$ which implies $b\not \in \mathbb{Z}$. Is this right? Am I missing anything?