Problem 837

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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 $A_3$?

Status. Open.

Source. erdosproblems.com/837, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #837, https://www.erdosproblems.com/837.

Formalization. Statement in formal-conjectures, added on 2026-10-07. The file states erdos_837 as research open, with the set $A_3$ left as an answer(sorry) and no formal_proof pointer, so no claim follows from it.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1979_problems_results_graph_theory_combinatorial_analysis
• erdos_1979_problems_results_graph_theory_combinatorial_analysis / conjecture_p158
• komorech_2025_non_jumping_densities_3_uniform_hypergraphs
• komorech_2025_non_jumping_densities_3_uniform_hypergraphs / proposition_1_4
• komorech_2025_non_jumping_densities_3_uniform_hypergraphs / theorem_1_5
• komorech_2025_non_jumping_densities_3_uniform_hypergraphs / theorem_1_6
• komorech_2025_non_jumping_densities_3_uniform_hypergraphs / theorem_4_8
• liu_2026_number_4_9_is_non_jump