Problem 1075

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claims/: The 1 claim page of Problem 1075, one per claimant's result; the problem's standing derives from them.

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Statement. Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds.

Any $r$-uniform hypergraph on $n$ vertices with at least $(1+\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\to \infty$ as $n\to \infty$.

Status. Open on the site (label OPEN; page last edited 5 October 2025). The site's commentary records Erdős's theorem [Er64f] that the statement holds with $c_r=r^{-r}$ for every hypergraph with at least $\epsilon n^r$ edges, so the question is whether the constant can be raised above $r^{-r}$ under the stronger density hypothesis. One full claim is pending: Gu's disproof (5 September 2026), explicit $5$-uniform hypergraphs meant to show that no $c_5>5^{-5}$ works, lifted to every $r\ge5$; the forum entry described an earlier $16$-uniform version, whose Zenodo record was removed on 23 September 2026. It is not refereed and no outside reviewer has endorsed it, so the standing is claimed, not solved.

Source. erdosproblems.com/1075, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1075, https://www.erdosproblems.com/1075.

References.

• [Er64f] Erdős, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183-190.
• [Er74c] Erdős, Paul, [[../library/extremal_graph_theory/erdos_1974_extremal_problems_graphs_hypergraphs/_index|Extremal problems on graphs and hypergraphs]]. (1974), 75-84.

Formalization. A formal-conjectures statement file, FormalConjectures/ErdosProblems/1075.lean, states the problem; the claimant's own Lean archive is linked from the claim page.

Progress

Not yet compiled.

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_1964_extremal_problems_graphs_generalized_graphs
• erdos_1974_extremal_problems_graphs_hypergraphs