Erdős (Erdos) Problem #766: status in 2026

Erdős Problem #766

Let f(n;k,l)=min ex(n;G), where G ranges over all graphs with k vertices and l edges. Give good estimates for f(n;k,l) in the range k<l≤ k^2/4. In [Er64c, p.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #766 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #766 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	4 open, 1 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.

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Cite

@misc{cairn-erdos-766,
  title        = {Erdős Problem #766},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-766}},
  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

erdos_766.parts.i. Let $f (n; k, l) = min ex (n; G)$ , where $G$ ranges over all graphs with $k$ vertices and $l$ edges.

Give good estimates for $f (n; k, l)$ in the range $k < l \leq k^{2} /4$ .

In [Er64c, p. 33] this question is asked for $f_{1} (n; k, l)$ (Erdos766.forcingNumber), which forces at least one graph with $k$ vertices and $l$ edges rather than one fixed graph; we follow the source. The question for the function of erdosproblems.com is Erdos766.erdos_766.variants.minExtremalNumber_estimate.

erdos_766.parts.ii. For fixed $k$ and large $n$ is $f (n; k, l)$ a strictly monotone function of $l$ ?

As in Erdos766.erdos_766.parts.i, $f$ is $f_{1}$ of [Er64c], and we use the range $k < l \leq k^{2} /4$ in which the question is asked there; here k ^ 2 / 4 is $⌊ k^{2} /4 ⌋$ . The question for the function of erdosproblems.com is Erdos766.erdos_766.variants.minExtremalNumber_strictMonoOn.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«766» (2 statements).

theorem erdos_766.parts.i :
    ∀ k l : ℕ, k < l → l ≤ k ^ 2 / 4 →
      (fun n ↦ (forcingNumber n k l : ℝ)) =Θ[atTop] erdos_766.parts.i.answer k l

theorem erdos_766.parts.ii :
    ∀ k : ℕ, ∀ᶠ n : ℕ in atTop,
      StrictMonoOn (forcingNumber n k) (Set.Ioc k (k ^ 2 / 4))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/766. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_766.variants.minExtremalNumber_estimate — The estimate question of Erdos766.erdos_766.parts.i for the function min ex(n;G) of erdosproblems.com (Erdos766.minExtremalNumber).
erdos_766.variants.minExtremalNumber_strictMonoOn — The monotonicity question of Erdos766.erdos_766.parts.ii for the function min ex(n;G) of erdosproblems.com (Erdos766.minExtremalNumber).

References

erdosproblems.com/766
[Er64c] Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. PDF

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.