# Has this kind of design been studied before?

Consider a design $(X,\mathcal{B})$, satisfying:

1. Each block in $\mathcal{B}$ has the same size

2. The intersection of every two blocks has the same size

Of course, it is easy to find many examples of such design. For instance, each symmetric balanced incomplete block design (BIBD) satisfies the above two conditions.

I am interested in the following question:

Given the number of points and the size of blocks, find the maximum number of blocks and construct the design achieving this maximum number.

I tried my best to check the 'Handbook of Combinatorial Designs' and found nothing. I am not sure if this kind of design has been studied before.

Under your hypotheses, the Frankl-Wilson theorem says that $|\mathcal{B}| \leq |X|$. More generally, if there are $s$ possible intersection sizes, $|\mathcal{B}| \leq \binom{|X|}{s}$.