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 $t+1 \in [\ell^2]$, there exist $3$ *disjoint* subsets of $S_{t+1}$: $F_1, F_2, F_3$ (these are "forbidden" subsets), such that:
   1. $|F_1|, |F_2|, |F_3| \ge \beta \cdot \ell$.
   2. $S_1, \dots, S_t$ 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 chunk $j \in [3]$, its intersection with $F_j$ is empty. In other words, for the first chunk,
$\left( \bigcup_{j=1}^{i_1} S_j \right) \cap F_1 = \emptyset$, and so on.

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