Erdős (Erdos) Problem #627: status in 2026

Erdős Problem #627

Let ω(G) denote the clique number of G and χ(G) the chromatic number. If f(n) is the maximum value of χ(G)/ω(G), as G ranges over all graphs on n vertices, then does lim_n→inftyf(n)/n/(log_2n)^2 exist?

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 #627 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_627 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #627 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_627`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 8 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-627,
  title        = {Erdős Problem #627},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-627}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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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 $ω (G)$ denote the clique number of $G$ and $χ (G)$ the chromatic number. If $f (n)$ is the maximum value of $χ (G) / ω (G)$ , as $G$ ranges over all graphs on $n$ vertices, then does $n \to \infty lim \frac{f ( n )}{n / ( lo g _{2} n ) ^{2}}$ exist?

Since $f (n) ≍ n / (lo g_{2} n)^{2}$ (see Erdos627.erdos_627.variants.theta), the limit, if it exists, is a finite real number.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«627».

theorem erdos_627 :
    ∃ L : ℝ, Tendsto (fun n : ℕ ↦ maxChromaticCliqueRatio n / ((n : ℝ) / Real.logb 2 n ^ 2))
      atTop (𝓝 L)

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/627. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/627
[AFM25] Araujo, I. and Filipe, R. and Miyazaki, R., *A note on the maximum ratio between chromatic number and clique number*. arXiv:2512.16062 (2025).
[Er61d] Erdős, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352.
[Er67c] Erdős, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256.
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
[My55] Mycielski, J., Sur le coloriage des graphs. Colloquium Mathematicum 3 (1955), 161–162. doi:10.4064/cm-3-2-161-162
[Zy52] Zykov, A. A., On some properties of linear complexes. Amer. Math. Soc. Translation (1952), 33.

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.