Erdős (Erdos) Problem #1013: status in 2026

Erdős Problem #1013

Let h_3(k) be the minimal n such that there exists a triangle-free graph on n vertices with chromatic number k. Find an asymptotic for h_3(k), and also prove lim_k→ inftyh_3(k+1)/h_3(k)=1. This is the first part.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-10).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1013 as open (status updated 10 September 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1013 on erdosproblems.com	open 2025-09-10	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 3 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-1013,
  title        = {Erdős Problem #1013},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1013}},
  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_1013.parts.i. Let $h_{3} (k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$ . Find an asymptotic for $h_{3} (k)$ , and also prove $k \to \infty lim \frac{h _{3} ( k + 1 )}{h _{3} ( k )} = 1.$

This is the first part.

erdos_1013.parts.ii. Let $h_{3} (k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$ . Find an asymptotic for $h_{3} (k)$ , and also prove $k \to \infty lim \frac{h _{3} ( k + 1 )}{h _{3} ( k )} = 1.$

This is the second part.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1013» (2 statements).

theorem erdos_1013.parts.i :
    (fun k ↦ (h3 k : ℝ)) ~[atTop] erdos_1013.parts.i.answer

theorem erdos_1013.parts.ii :
    Tendsto (fun k : ℕ ↦ (h3 (k + 1) : ℝ) / h3 k) atTop (nhds 1)

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/1013. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1013
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[GrYa68] Graver, J. E. and Yackel, J., *Some graph theoretic results associated with Ramsey's theorem*. J. Combinatorial Theory 4 (1968), 125-175.
[DaIl22] Davies, E. and Illingworth, F., The $χ$ -Ramsey problem for triangle-free graphs. SIAM J. Discrete Math. 36 (2022), 1124-1134.
[HHKP25] Hefty, Z., Horn, P., King, D. and Pfender, F., Improving $R (3, k)$ in just two bites. arXiv:2510.19718 (2025).

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.