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Level A · Machine-checkable Hard Graph theory P-erdos-609

Erdős Problem #609

Let f(n) be the minimal m such that if the edges of K_2^n+1 are coloured with n colours then there must be a monochromatic odd cycle of length at most m. Estimate f(n).

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 #609 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_609 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #609 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_609`research openFormal Conjectures
Formal Conjectures: research statements in the file1 open, 0 solvedFormal 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-609,
  title        = {Erdős Problem #609},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-609}},
  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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On the literature board
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Nobody has worked on this problem here yet

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  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

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.

or , ,

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let be the minimal such that if the edges of are coloured with colours then there must be a monochromatic odd cycle of length at most . Estimate .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«609».

theorem erdos_609 :
    (fun n ↦ (f n : ℝ)) =Θ[atTop] erdos_609.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/609. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/609
  • [ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Colloq. Math. Soc. János Bolyai (1975).
  • [Ch97] Chung, F., Open problems of Paul Erdős in graph theory. J. Graph Theory (1997), 3-36.
  • [DaJo17] Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.
  • [GiHu24] Girão, A. and Hunter, Z., Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).
  • [JaYi25] Janzer, O. and Yip, F., Short monochromatic odd cycles. arXiv:2506.14910 (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.