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on Erdős #837 · graph theory / hypergraphs

The statement

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 $A_3$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.