## Existence of a block design

Let $\ell$ be an integer parameter. I want to ask the existence of the following design: There is a universal constant $\beta < 1$ such that for all sufficiently large $\ell$, the following holds: The universe size $d = O(\ell)$. There are $\ell^2$ sets in the design, $S_1, \dots, S_{\ell^2} \subseteq [d]$. $(\forall i \in … Read more