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 $\mathrm{\mid }A\cap B\mathrm{\mid }\ge t$ 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]}$\mathrm{.}Aconstructionfor$3$-laminar families and a crude analysis for general $t$ are also given.