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, $S_1, \dots, S_{\ell^2} \subseteq [d]$.
 - $(\forall i \in [\ell^2])\ |S_i| = \ell$.
 - For every $t+1 \in [\ell^2]$, look at the $t$ sets $S_1, \dots, S_t$ before $S_{t+1}$, then these sets can be divided into $3$ consecutive chunks, $S_1, \dots, S_{i_1}$; $S_{i_1+1}, \dots, S_{i_2}$; $S_{i_2+1}, \dots, S_t$ such that for each chunk, its intersection with $S_{t+1}$ leaves $\beta \cdot \ell$ elements in $S_{t+1}$ uncovered. In other words, for the first chunk, $\left|\left(\bigcup_{j=1}^{i_1} S_j \right) \cap S_{t+1}\right| \le (1-\beta) \cdot \ell$.

btw. do we have a name for such block designs?