Erdős (Erdos) Problem #1182: status in 2026

Erdős Problem #1182

Let f(n) be maximal such that there is a connected graph G with n vertices and f(n) edges such that R(K_3,G)= 2n-1. Let F(n) be maximal such that every connected graph G with n vertices and ≤ F(n) edges has R(K_3,G)= 2n-1. Estimate f(n) and F(n). In particular, is it true that F(n)/n→ ∞?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-03-07).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1182 as open (status updated 7 March 2026).
Formal Conjectures marks the Lean statement erdos_1182.parts.i as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1182 on erdosproblems.com	open 2026-03-07	erdosproblems.com
Formal Conjectures: status of `erdos_1182.parts.i`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 11 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-1182,
  title        = {Erdős Problem #1182},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1182}},
  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

Nobody has worked on this problem here yet

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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

Let $f (n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f (n)$ edges such that $R (K_{3}, G) = 2 n - 1.$ Let $F (n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\leq F (n)$ edges has $R (K_{3}, G) = 2 n - 1.$ Estimate $f (n)$ and $F (n)$ . In particular, is it true that $F (n) / n \to \infty$ ?

This is the first part: estimate $f (n)$ . The problem is due to Burr, Erdős, Faudree, Rousseau and Schelp [BEFRS80] and appears in [Er78, p. 33]. We read "estimate" as the order of magnitude =Θ. Burr, Erdős, Faudree, Rousseau and Schelp [BEFRS80, Theorem 2] proved $n^{3/2} (lo g n)^{1/2} ≪ f (n) ≪ n^{5/3} (lo g n)^{2/3}$ (erdos_1182.variants.f_lower, erdos_1182.variants.f_upper).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1182».

theorem erdos_1182.parts.i :
    (fun n : ℕ ↦ (f n : ℝ)) =Θ[atTop] erdos_1182.parts.i.answer

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/1182. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_1182.variants.brandt_expects — Brandt [Br96] expects that for all large n we have 2n<F(n)<6n (as recorded on the problem page).

References

erdosproblems.com/1182
[Er78] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number theory*. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. PDF
[BEFRS80] Burr, S. A., Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H., *An extremal problem in generalized Ramsey theory*. Ars Combin. 10 (1980), 193-203. PDF
[Br96] Brandt, S., Expanding graphs and Ramsey numbers. Freie Universität Berlin, preprint A 96-24 (1996).
[BBH98] Brandt, S., Brinkmann, G. and Harmuth, T., *All Ramsey numbers $r (K_{3}, G)$ for connected graphs of order 9*. Electron. J. Combin. 5 (1998), #R7.
[Ch77] Chvátal, V., Tree-complete graph Ramsey numbers. J. Graph Theory 1 (1977), 93.
[NiRo09] Nikiforov, V. and Rousseau, C. C., Ramsey goodness and beyond. Combinatorica 29 (2009), 227-262.

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.