Erdős (Erdos) Problem #1178: status in 2026

Erdős Problem #1178

For r≥ 3 let d_r(e) be the minimal d such that ex_r(n,F)=o(n^2), where F is the family of r-uniform hypergraphs on d vertices with e edges. Prove that d_r(e)=(r-2)e+3 for all r,e≥ 3.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-01-25).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1178 as open (status updated 25 January 2026).
Formal Conjectures marks the Lean statement erdos_1178 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1178 on erdosproblems.com	open 2026-01-25	erdosproblems.com
Formal Conjectures: status of `erdos_1178`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-1178,
  title        = {Erdős Problem #1178},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1178}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

For $r \geq 3$ let $d_{r} (e)$ be the minimal $d$ such that $ex_{r} (n, F) = o (n^{2}),$ where $F$ is the family of $r$ -uniform hypergraphs on $d$ vertices with $e$ edges. Prove that $d_{r} (e) = (r - 2) e + 3$ for all $r, e \geq 3$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1178».

theorem erdos_1178 :
    ∀ r e : ℕ, 3 ≤ r → 3 ≤ e →
    Hypergraph.sparseConfigurationThreshold r e = (((r - 2) * e + 3 : ℕ) : ℕ∞)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1178. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1178
[BES73] Brown, W. G. and Erdős, P. and S\'os, V. T., Some extremal problems on { $r$ }-graphs. (1973), 53--63.
[CGLS23] Conlon, David and Gishboliner, Lior and Levanzov, Yevgeny and Shapira, Asaf, A new bound for the {B}rown-{E}rd\H os-S\'os problem. J. Combin. Theory Ser. B (2023), 1--35.
[EFR86] Erdős, P. and Frankl, P. and Rödl, V., The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. (1986), 113-121.
[Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journ\'{e}es Arithm\'{e}tiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.
[RuSz78] Ruzsa, I. Z. and Szemer\'{e}di, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (1978), 939-945.
[SaSe05] Sárk\"ozy, Gábor N. and Selkow, Stanley, An extension of the {R}uzsa-Szemer\'edi theorem . Combinatorica (2005), 77--84.
[SoSo17] Solymosi, David and Solymosi, Jozsef, Small cores in 3-uniform hypergraphs. J. Combin. Theory Ser. B (2017), 897--910.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.