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 $A_3$?
