Erdős (Erdos) Problem #626: status in 2026

Erdős Problem #626

Let k ≥ 4 and let g_k(n) denote the largest m such that there is a graph on n vertices with chromatic number k and girth > m (i.e. containing no cycle of length ≤ m). Does lim_n→inftyg_k(n)/log n 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 #626 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #626 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	4 open, 4 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-626,
  title        = {Erdős Problem #626},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-626}},
  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_626.parts.i. Let $k \geq 4$ and let $g_{k} (n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $> m$ (i.e. containing no cycle of length $\leq m$ ). Does $n \to \infty lim \frac{g _{k} ( n )}{lo g n}$ exist?

By Erdos626.erdos_626.variants.kostochka and Erdos626.erdos_626.variants.erdos_upper, the limit, if it exists, is a positive real number.

erdos_626.parts.ii. Let $h^{(m)} (n)$ be the maximal chromatic number of a graph on $n$ vertices with girth $> m$ . Does $n \to \infty lim \frac{lo g h ^{(m)} ( n )}{lo g n}$ exist, and what is its value? The value is asked for in Erdos626.erdos_626.variants.h_odd_conjecture and Erdos626.erdos_626.variants.h_even_conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«626» (2 statements).

theorem erdos_626.parts.i :
    ∀ k : ℕ, 4 ≤ k → ∃ L : ℝ,
      Tendsto (fun n : ℕ ↦ (g k n : ℝ) / Real.log n) atTop (𝓝 L)

theorem erdos_626.parts.ii :
    ∀ m : ℕ, ∃ L : ℝ,
      Tendsto (fun n : ℕ ↦ Real.log (h m n : ℝ) / Real.log n) atTop (𝓝 L)

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/626. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_626.variants.h_odd_conjecture — Erdős conjectured that the bound in Erdos626.erdos_626.variants.h_upper_odd is sharp: for odd m, lim_n→∞log h^(m)(n)/log n = 2/m+1.
erdos_626.variants.h_even_conjecture — For even m, Erdős had no good guess for the value of the limit, other than that it should lie in [2/m+2, 2/m]. He could not prove this even for m = 4.

References

erdosproblems.com/626
[Er59b] Erdős, P., Graph theory and probability. Canadian J. Math. (1959), 34-38.
[Er62b] Erdős, P., On circuits and subgraphs of chromatic graphs. Mathematika (1962), 170-175.
[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.
[Ko88] Kostochka, A. V., Upper bounds on the chromatic number of graphs. Trudy Inst. Mat. (Novosibirsk) (1988), 204-226, 265.

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.