Erdős Problem #569 — open problem in graph theory

Erdős Problem #569

Let k≥ 1. What is the best possible c_k such that R(C_2k+1,H)≤ c_k m for any graph H on m edges without isolated vertices? This problem is #34 in Ramsey Theory in the graphs problem collection.

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-569,
  title        = {Erdős Problem #569},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-569}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $k \geq 1$ . What is the best possible $c_{k}$ such that $R (C_{2 k + 1}, H) \leq c_{k} m$ for any graph $H$ on $m$ edges without isolated vertices?

This problem is #34 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«569». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_569 :
    let c : ℕ → ℝ := answer(sorry)
    ∀ (k : ℕ) (hk : 1 ≤ k),
      sInf {C : ℝ | 0 < C ∧
        ∀ (m : ℕ) (W : Type) [Fintype W] (H : SimpleGraph W) [DecidableRel H.Adj],
          (∀ v, 0 < H.degree v) →
          H.edgeSet.ncard = m →
          (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph (2 * k + 1)) H : ℝ) ≤ C * m} =
            c k

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/569. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/569
[EFRS93] Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.

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.