Erdős Problem #1022 — Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are…

Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t\lvert X\rvert$ many $A\in \mathcal{F}$ with $A\subseteq X$, then $\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?