Erdős (Erdos) Problem #554: status in 2026

Erdős Problem #554

Let R_k(G) denote the minimal m such that if the edges of K_m are k-coloured then there is a monochromatic copy of G. Show that lim_k→ inftyR_k(C_2n+1)/R_k(K_3)=0 for any n≥ 2. A problem of Erdős and Graham. The case n = 1 is excluded since C_3 = K_3.

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 #554 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_554 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #554 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_554`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 5 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-554,
  title        = {Erdős Problem #554},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-554}},
  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

Let $R_{k} (G)$ denote the minimal $m$ such that if the edges of $K_{m}$ are $k$ -coloured then there is a monochromatic copy of $G$ . Show that $k \to \infty lim \frac{R _{k} ( C _{2 n + 1} )}{R _{k} ( K _{3} )} = 0$ for any $n \geq 2$ .

A problem of Erdős and Graham. The case $n = 1$ is excluded since $C_{3} = K_{3}$ . This problem is #23 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«554».

theorem erdos_554 (n : ℕ) (hn : 2 ≤ n) :
    Tendsto (fun k : ℕ ↦ (multicolourGraphRamsey k (cycleGraph (2 * n + 1)) : ℝ) /
      (multicolourGraphRamsey k (completeGraph (Fin 3)) : ℝ)) atTop (𝓝 0)

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/554. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_554.variants.n_eq_two — The problem is open even for n = 2: lim_k→ inftyR_k(C_5)/R_k(K_3)=0.

References

erdosproblems.com/554
[Er81c] Erdős, P., *Some new problems and results in graph theory and other branches of combinatorial mathematics*. Combinatorics and graph theory (Calcutta, 1980), Lecture Notes in Math. 885 (1981), 9-17.
[BoEr73] Bondy, J. A. and Erdős, P., Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 (1973), 46-54.
[ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (Colloq., Keszthely, 1973), Colloq. Math. Soc. János Bolyai 10 (1975), 515-527.
[DaJo17] Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B 124 (2017), 56-63. arXiv:1602.07607
[JeSk21] Jenssen, M. and Skokan, J., *Exact Ramsey numbers of odd cycles via nonlinear optimisation*. Adv. Math. 376 (2021), Paper No. 107444. arXiv:1608.05705
[ACJMR26] Axenovich, M., Cames van Batenburg, W., Janzer, O., Michel, L. and Rundström, M., An improved upper bound for the multicolour Ramsey number of odd cycles. J. Combin. Theory Ser. B 179 (2026), 293-298. arXiv:2510.17981 doi:10.1016/j.jctb.2026.04.005

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.