Generalized laminar families and certain forbidden matrices

Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system $\mathcal{F} \subseteq 2^X$ is $t$-laminar if $A,B \in \mathcal{F}$ with $|A \cap B| \ge t$ implies $A \subseteq B$ or $B \subseteq A$. We obtain very close asymptotic bounds in terms of $n$ on the maximum size of a $2$-laminar family $\mathcal{F} \subseteq 2^{[n]}$. A construction for $3$-laminar families and a crude analysis for general $t$ are also given.