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Hindman’s finite sums and products conjecture

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

Proves Hindman’s finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

It bears on Erdős problem #172, as listed by Thomas Bloom, erdosproblems.com.

What the papers state

Theorem 1.1

In Monochromatic finite sums and products in the positive integers, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let r,m\ge1 be integers, and fix real numbers R\ge2 and D\ge1. For every coloring \chi:\N\to[r], there are distinct positive integers a_1<\cdots<a_m and a color c\in[r] such that

\chi\left(\sum_{j\in J}a_j\right) =\chi\left(\prod_{j\in J}a_j\right)=c \quad\text{for every }\varnothing\ne J\subseteq[m].

Equivalently, \FS(A)\cup\FP(A) is monochromatic for a set A\subset\N of cardinality m. The elements can additionally be chosen to satisfy

a_1>R,\qquad a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D \quad(2\le d\le m).

Corollary 1.2

In Monochromatic finite sums and products in the positive integers, 23 September 2026

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

For every finite coloring of \N and every m\ge1, there is an m-element set A\subset\N for which \FS(A)\cup\FP(A) is monochromatic and

|\FS(A)|=|\FP(A)|=2^m-1,\qquad \FS(A)\cap\FP(A)=A.

In particular, the union has 2(2^m-1)-m distinct elements.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

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Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Other reports of proof verification

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

  • Yao Xu, feat(ErdosProblems/172): mark solved and link the formal proof (PR 7127, fixes issue 7126), 11 October 2026. Source

    Reported checks: Comparator and nanoda against a Mathlib-only challenge file and a fresh-kernel replay of the full dependency cone, with axioms propext, Classical.choice, Quot.sound.

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.

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

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

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

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • Yao Xu, feat(ErdosProblems/172): mark solved and link the formal proof (PR 7127, fixes issue 7126), 11 October 2026. Source

    Erdős 172 marked solved with a formal_proof link to hindman_finite_sums_products via a derived right-hand-side theorem.

  • The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source

    JSP-000168 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: No.

  • Yao Xu, Hindman sums and products in Lean 4 (following OpenAI’s proof), 7 October 2026. Source

    Lean proof of Hindman sums and products for all finite colourings, with the three standard axioms.

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

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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

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

    Bloom lists Erdős 172 (Hindman’s conjecture on sums and products), notes Alweiss proved it over the rationals, and says it is not formalised and the PDF has 70 pages.

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.

Identify the theorem the catalogue describes

  • Family 164 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 164 The release has no Lean statement for this family. Open the form