Number of blocks that meet a particular block in a 2 design

Question

Let D be a $2-(v, k, λ)$ design with b blocks and r blocks through every point. Let B be any block. How to show that the number of blocks that meet B is at least $k(r − 1)^2 /[(k − 1)(λ − 1) + (r − 1)]$. Moreover when does equality hold? [This is a problem in Lint, Wilson`s book A course in combinatorics page 224] Try: let $a_i$ be the number of blocks different from B which meets b at exactly i points. Then I have found expressions for $\sum a_i$, $\sum ia_i$, $\sum i(i-1)a_i$, as given in hints. After that how to proceed?

Answer 1

You can calculte $\sum\limits_{i=1}i^2a_i$ , and apply Cauchy inequality to get an lower bound of $\sum\limits_{i=1} a_i$ which is the number of blocks that meet B. And the value of it is $k(r−1)^2/[(k−1)(λ−1)+(r−1)]$