Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need \(O_F(\Delta/\log\Delta)\) colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size \(\Omega_r(n\log d/d)\). (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.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments
It bears on Erdős problem #802, as listed by Thomas Bloom, erdosproblems.com.
In A logarithmic independence bound for clique-free graphs, 25 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer r\ge4, there is a constant c_r>0 such that every finite simple K_r-free graph G with n vertices and average degree d\ge2 satisfies
In Correspondence coloring graphs with a forbidden clique, 5 October 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer r\ge4 there are constants C_r>0 and \Delta_r\ge3 such that every K_r-free graph G of maximum degree \Delta\ge\Delta_r satisfies
The constants depend only on r.
No Lean statement in the release is paired with this theorem.
In Correspondence coloring graphs with a forbidden clique, 5 October 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every finite graph F there are constants C_F>0 and \Delta_F\ge3 such that every F-free graph G of maximum degree \Delta\ge\Delta_F satisfies
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.
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.
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
CliqueFreeLogThe paper states
In A logarithmic independence bound for clique-free graphs, 25 September 2026
For every integer r\ge4, there is a constant c_r>0 such that every finite simple K_r-free graph G with n vertices and average degree d\ge2 satisfies
The words they share: average, simple, integer, degree, free, finite, log, graph, logarithmic, independence, clique, bound, vertices, constant.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CliqueFreeLog.lean, the release lists it as a main result.
theorem logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) : ∃ c : ℝ, 0 < c ∧ ∀ {V : Type} [Fintype V] (G : SimpleGraph V), G.CliqueFree r → 2 ≤ averageDegree G → c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤ (G.indepNum : ℝ) := 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Only the independence half is stated: logarithmic_independence_bound (r) (hr : 4 ≤ r) : ∃ c > 0, ∀ G, G.CliqueFree r → 2 ≤ averageDegree G → c * n * log d / d ≤ G.indepNum (averageDegree = 2|E|/|V|). The summary’s first and primary claim, the Alon-Krivelevich-Sudakov coloring conjecture in correspondence-coloring form (O_F(Δ/log Δ) colors for any fixed forbidden subgraph F), has no Lean statement, and the docs scope also only mentions the independence bound.
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name: The correspondence-coloring theorem (χ_DP = O_F(Δ/log Δ) for F-free graphs of large maximum degree), the headline’s first clause, is not stated.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
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 formalized result gives a logarithmic improvement in the independence number of clique-free graphs. For every fixed integer $r\ge4$, there is $c_r>0$ such that every finite $K_r$-free simple graph on $n$ vertices with average degree $d\ge2$ has an independent set of size at least $c_r n\log d/d$. The constant depends only on $r$.
Logarithmic independence bound for clique-free graphs CliqueFreeLog
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for each r ≥ 4, every K_r-free graph with average degree d ≥ 2 has an independent set of size at least c_r n log d/d. The correspondence-colouring result of the family’s later paper is not part of the statement.
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.
CliqueFreeLogThree reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original 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 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 358 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
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.
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-000659 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
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
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 the Lean statement with the paperSuggest a source attributionSuggest a change
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 “Major math journal”. 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 802 (independent sets in clique-free graphs), names Shearer and Ajtai, Erdős, Komlós and Szemerédi, 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 A logarithmic independence bound for clique-free graphs, 25 September 2026
Why this passage is shown
Theorem 1.1
In Correspondence coloring graphs with a forbidden clique, 5 October 2026
Why this passage is shown
Corollary 1.2
In Correspondence coloring graphs with a forbidden clique, 5 October 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CliqueFreeLog
Proof-verification report
MathVet, CliqueFreeLog
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, CliqueFreeLog
Lean statement
CliqueFreeLog