I'm looking to prove that the size of the intersect of two (v, k, λ)-designs cannot be equal to (v-1). I'm aware how to prove this for a (v,3,1) design as;
In a STS (v,3,1) the number of blocks is v and the replication number is 3. So if |B1 ∩ B2| = v-1 then they differ in 1 block. But without loss of generality this means that {1, 2, 3} ∈ B1 while {1, 2, 3} ̸∈ B2. But since the replication number is 3 and all other blocks are the same we need a block in B2 which contains 1,2 and 3 once each. The only possibility is {1, 2, 3}; a contradiction.
I was wondering if this could be extended to a (v, k, λ)-design or if I should be taking a different approach.
Thanks in advance :)