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Sharp logarithmic exponents for off-diagonal Ramsey numbers

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

For every fixed integer s ≥ 5, proves \(r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}\) as \(t\to\infty\), determining the logarithmic exponent and matching the classical upper bound at that scale. Here \(r(s,t)\) is the least number of vertices forcing an s-clique or a t-vertex independent set. (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 #986, as listed by Thomas Bloom, erdosproblems.com.

What the papers state

Theorem 1.1

In Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every fixed integer s\ge6 there is a constant C_s>0 such that, for every \varepsilon>0 and every sufficiently large integer t,

\frac{t^{s-1}}{(\log t)^{s-2+\varepsilon}} \le r(s,t) \le C_s\frac{t^{s-1}}{(\log t)^{s-2}}.

In particular,

\lim_{t\to\infty} \frac{(s-1)\log t-\log r(s,t)}{\log\log t}=s-2.

Theorem 1.2

In Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Fix an integer d\ge5 and a real number 0<\eta<1/10. For every sufficiently large prime q, with \sigma=\log q, there is a graph G such that

|V(G)|=\lfloor q^d\sigma\rfloor,\qquad K_{d+1}\not\subseteq G,\qquad \alpha(G)<\lfloor q\sigma^{1+\eta}\rfloor.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

One more possible match

  • Theorem 1.1 in The sharp logarithmic exponent of r(5,t), 24 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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.

SharpLogRamsey

The paper states

The words they share: constant, fixed, integer.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Sharp logarithmic exponent for fixed off-diagonal Ramsey numbers

SharpLogRamsey.lean, the release lists it as a supporting statement.

theorem main (s : ℕ) (hs : 6 ≤ s) : MainBounds s ∧ MainLimit s := by  sorry

This 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.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

RamseyFive

The paper states

No paper theorem is paired with this Lean statement.

no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements it shares words with:

  • Theorem 1.1 in Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026.
  • Theorem 1.1 in The sharp logarithmic exponent of r(5,t), 24 September 2026.
  • Proposition 7.3 (Upper Ramsey estimate) in Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026.
  • Proposition 3.5 (Low-dimensional preparation) in Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026.
  • Theorem 1.2 in Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026.
  • Theorem 2.1 in The sharp logarithmic exponent of r(5,t), 24 September 2026.
  • Proposition 6.4 (One compression stage) in The sharp logarithmic exponent of r(5,t), 24 September 2026.
  • Proposition 3.4 (Repeated projection) in Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers, 24 September 2026.

The Lean statement

No paper theorem is paired with it.

Sharp logarithmic exponent of $r(5,t)$

RamseyFive.lean, the release lists it as a supporting statement.

theorem main : SharpBounds ∧ SharpExponent := by  sorry

This 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.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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: RamseyFive.main : SharpBounds ∧ SharpExponent covers s=5 (lower t^4/(log t)^(3+ε), upper C t^4/(log t)^3, and (4 log t - log r(5,t))/log log t -> 3); SharpLogRamsey.main (s) (hs : 6 ≤ s) : MainBounds s ∧ MainLimit s covers every s >= 6, so together all s >= 5. r(s,t) = sInf {n | ∀ G : SimpleGraph (Fin n), K_s ∨ independent t-set} with Mathlib IsNClique/IsNIndepSet, standard.

“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.

Read the original assessment Read MathVet’s method

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.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number $r(5,t)$. It proves $r(5,t)=t^4/(\log t)^{3+o(1)}$ as $t\to\infty$. For every $\varepsilon>0$, the lower bound $t^4/(\log t)^{3+\varepsilon}$ holds eventually, while the upper bound is $Ct^4/(\log t)^3$ for an absolute $C>0$. The corresponding logarithmic exponent converges to three along all natural values of $t$.

The formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number for every fixed integer $s\ge6$. It proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ as $t\to\infty$. More explicitly, for each $\varepsilon>0$ the lower bound $t^{s-1}/(\log t)^{s-2+\varepsilon}$ holds eventually, while the upper bound is $C_s t^{s-1}/(\log t)^{s-2}$. The logarithmic exponent limit is taken along all natural values of $t$.

AI-drafted comparisons, with no recorded human review

  • Sharp logarithmic exponent for fixed off-diagonal Ramsey numbers SharpLogRamsey

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every fixed s ≥ 6, r(s, t) = t^{s−1}/(log t)^{s−2+o(1)}, with upper bound C_s t^{s−1}/(log t)^{s−2}.

  • Sharp logarithmic exponent of $r(5,t)$ RamseyFive

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: r(5, t) = t⁴/(log t)^{3+o(1)}, with upper bound Ct⁴/(log t)³.

Who ran the proof-verification program?

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.

Sharp logarithmic exponent for fixed off-diagonal Ramsey numbers SharpLogRamsey

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Sharp logarithmic exponent of $r(5,t)$ RamseyFive

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Other reports of proof verification

Other people’s accounts of running proof-verification software, with links to their reports.

  • acprk, Remarks on proofs in OpenAI’s mathematics repository, 10 October 2026. Source

    The release’s existing Lean proof already covers the case s = 4 of the off-diagonal Ramsey exponent, checked by Lean kernel replay (main_four).

What these papers cite in the release, and what cites them

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.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

References outside the release

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

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare one Lean statement with its paper: RamseyFive, SharpLogRamseySuggest a source attributionSuggest a change

What people have said

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 “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Compiler not recorded (Atlas survey), Atlas survey of Erdős problem pages, recorded on 11 October 2026, 11 October 2026. Source

    The imported Atlas survey reports: Problem 986 page credits an internal OpenAI model with the adjustment that improved a lower bound exponent from s-2 to s-1 in Bradac’s work, last edited 21 Jun 2026; it does not cite the release’s papers.

  • MalekZ, jleng01, NicoV, efiebig, Thomas Bloom, Comments on ‘Erdős problems in the OpenAI proofs’, 9 October 2026. Source

    A commenter notes that if the off-diagonal Ramsey manuscripts are correct, Erdős problem 986 benefits and by extension several related problems, which the site maintainer then recorded.

  • Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source

    Bloom lists Erdős 986 (off-diagonal Ramsey numbers) and notes that Bradač had already answered the original question earlier in the year, so the claim improves the logarithmic power.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 170 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Compare the proposed paper and Lean matches

  • Family 170 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form