Erdős Problem #837

declared status 'open'. Formalized: no. 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 $A_3$? Current best: It is known that$$A_2 = \left\{ 1-\frac{1}{k} : k\geq 1\right\}.$$ Prize: no. Tags: graph theory, hypergraphs.

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