Erdős Problem #837

Let \(k\geq 2\) and \(A_k\subseteq [0,1]\) be the set of \(\alpha\) such that there exists some \(\beta(\alpha)>\alpha\) with the property that, if \(G_1,G_2,\ldots\) is a sequence of \(k\)-uniform hypergraphs with \[\liminf \frac{e(G_n)}{\binom{\lvert G_n\rvert}{k}} >\alpha\] then there exist subgraphs \(H_n\subseteq G_n\) such that \(\lvert H_n\rvert \to \infty\) and \[\liminf \frac{e(H_n)}{\binom{\lvert H_n\rvert}{k}} >\beta,\] and further that this property does not necessarily hold if \(>\alpha\) is replaced by \(\geq \alpha\). What is \(A_3\)?