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)?

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)?

This is false, and $c_t<2$ for all $t$: a counterexample is provided by Wood [Wo13b], who constructs, for any $r\geq 2$, a triangle-free $2$-degenerate $r$-uniform hypergraph with chromatic number $3$. A similar counterexample was found independently by KoishiChan in the comments.

This was formalized in Lean by Alexeev using Aristotle.