Proves the Erdős–Faudree–Rousseau–Schelp conjecture: \(R(C_m,K_n)=(m-1)(n-1)+1\) for every \(m\ge n\ge3\), except \(R(C_3,K_3)=6\). This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
It bears on Erdős problem #551, as listed by Thomas Bloom, erdosproblems.com.
The paper states
In Cycle–clique Ramsey numbers, 25 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For all integers m\ge n\ge3 with (m,n)\ne(3,3),
For the excluded pair, R(C_3,K_3)=6.
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
CycleCliqueRamseyThe paper states
In Cycle–clique Ramsey numbers, 25 September 2026
For all integers m\ge n\ge3 with (m,n)\ne(3,3),
For the excluded pair, R(C_3,K_3)=6.
The words they share: integers, ramsey, cycle, clique.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CycleCliqueRamsey.lean, the release lists it as a supporting statement.
theorem thm_main : (∀ m n : ℤ, n ≤ m → 3 ≤ n → (m, n) ≠ (3, 3) → (cycleCliqueRamsey m.toNat n.toNat : ℤ) = (m - 1) * (n - 1) + 1) ∧ cycleCliqueRamsey 3 3 = 6 := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. Who ran the proof-verification program
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: thm_main : (∀ m n : ℤ, n ≤ m → 3 ≤ n → (m, n) ≠ (3, 3) → (cycleCliqueRamsey m.toNat n.toNat : ℤ) = (m - 1) * (n - 1) + 1) ∧ cycleCliqueRamsey 3 3 = 6, with RamseyProperty m n N := ∀ G : SimpleGraph (Fin N), cycleGraph m ⊑ G ∨ (⊤ : SimpleGraph (Fin n)) ⊑ Gᶜ (Mathlib IsContained) and sInf over ℕ: the exact EFRS statement. Note the import cone is 2.8M lines of lean/OAI, the largest in this range.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The cycle–clique Ramsey conjecture predicts the exact number of vertices forcing either a cycle $C_m$ or a clique $K_n$ in complementary colors. The formalization proves $R(C_m,K_n)=(m-1)(n-1)+1$ for all integers $m\ge n\ge3$ except $(m,n)=(3,3)$, where it proves $R(C_3,K_3)=6$. This is the complete parameter range of the paper’s main theorem.
Exact cycle–clique Ramsey numbers CycleCliqueRamsey
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem over its full range: R(C_m, K_n) = (m − 1)(n − 1) + 1 for m ≥ n ≥ 3 except (3, 3), where it is 6.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
CycleCliqueRamseyOne report, at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “REFUSED” | Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)). | 10 October 2026 | Original report |
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Proposition 9.1 of Cycle–clique Ramsey numbers, 25 September 2026
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000442 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
anxkhn, Series of 15 issues and 15 draft PRs adding release results to Formal Conjectures (issues 6972 to 6997, PRs 6977 to 7002), 8 October 2026. Source
Proposes adding the release’s full cycle-clique Ramsey formula (Erdős 551) to Formal Conjectures (issue 6987, PR 6992).
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Author not stated. Source
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 551 (the Ramsey numbers of cycles against complete graphs) and links the formalisation.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In Cycle--clique Ramsey numbers, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CycleCliqueRamsey
Report using another proof or calculation method
Proposition 9.1 of Cycle–clique Ramsey numbers, 25 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
CycleCliqueRamsey