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 \(<c_t\lvert X\rvert\) 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.