Erdős (Erdos) Problem #556: status in 2026

Erdős Problem #556

Erdős Problem 556 (Bondy–Erdős): for all n ≥ 4, R(C_n; 3) ≤ 4n - 3. The case n = 3 is excluded since R(K_3; 3) = 17.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). It is marked decidable there: it has been reduced to a finite computation.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #556 as decidable (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_556 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #556 on erdosproblems.com	decidable 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_556`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 5 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. 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-556,
  title        = {Erdős Problem #556},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-556}},
  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

Erdős Problem 556 (Bondy–Erdős): for all $n \geq 4$ , $R (C_{n}; 3) \leq 4 n - 3.$ The case $n = 3$ is excluded since $R (K_{3}; 3) = 17$ .

A problem of Bondy and Erdős: for the cycle $C_{n}$ with $n \geq 4$ , the $3$ -colour Ramsey number satisfies $R (C_{n}; 3) \leq 4 n - 3.$ The triangle is excluded: $C_{3} = K_{3}$ and $R (K_{3}; 3) = 17 > 9$ [Rad]. The bound is best possible for odd $n$ , where equality is conjectured (and known for large odd $n$ ). Łuczak proved the asymptotic bound $R (C_{n}; 3) \leq (4 + o (1)) n$ , and $(3 + o (1)) n$ for even $n$ ; Kohayakawa, Simonovits and Skokan settled large odd $n$ , and Benevides and Skokan large even $n$ , but the inequality for all $n$ remains open.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«556».

theorem erdos_556 (n : ℕ) (hn : 4 ≤ n) :
    multicolourRamsey (cycleGraph n) 3 ≤ 4 * n - 3

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/556. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/556
[Er81] P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25-42.
[Lu99] T. Łuczak, $R (C_{n}, C_{n}, C_{n}) \leq (4 + o (1)) n$ , J. Combin. Theory Ser. B 75 (1999), 174-187.
[KSS05] Y. Kohayakawa, M. Simonovits, J. Skokan, The 3-colored Ramsey number of odd cycles, Proceedings of GRACO2005 (2005), 397-402.
[BeSk09] F. S. Benevides, J. Skokan, The 3-colored Ramsey number of even cycles, J. Combin. Theory Ser. B 99 (2009), 690-708.
[Rad] S. P. Radziszowski, Small Ramsey Numbers, Electron. J. Combin. Dynamic Survey DS1, revision 18, p. 61.

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.