Erdős Problem #609 — open problem in combinatorics

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).

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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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 $f (n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«609». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_609 :
    (fun n ↦ (f n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)

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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.