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Hindman's finite sums and products conjecture

For finite A⊂NA\subset\mathbb N let FS(A)\mathrm{FS}(A) and FP(A)\mathrm{FP}(A) be the sums and products of its nonempty subsets. The Folkman-Rado-Sanders theorem gives monochromatic FS(A)\mathrm{FS}(A) for arbitrarily large AA in every finite coloring, and the multiplicative analogue follows, but the two together are different: Hindman (1980) found a finite coloring with no infinite set whose sums and products are monochromatic, and even the pattern {x,y,x+y,xy}\{x,y,x+y,xy\} was open for colorings of N\mathbb N in more than two colors (Moreira proved {x,x+y,xy}\{x,x+y,xy\}; Alweiss proved the full statement over Q\mathbb Q). Hindman (1979) conjectured the finite version. For every finite coloring of N\mathbb N and every mm, is there an mm-element set AA with FS(A)∪FP(A)\mathrm{FS}(A)\cup\mathrm{FP}(A) monochromatic?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Arithmetic Ramsey theory
Posed by
Neil Hindman (Trans. Amer. Math. Soc. 247, 1979); also Hindman-Strauss, Question 17.18
Year posed
1979
Years open
47y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every finite coloring of N\mathbb N and every m≥1m\ge1 there are a1<⋯<ama_1<\dots<a_m such that every nonempty subset sum and subset product has one common color; the ada_d can be made to grow faster than any prescribed power of the earlier sums and products, so all 2(2m−1)−m2(2^m-1)-m expressions are distinct. In particular {x,y,x+y,xy}\{x,y,x+y,xy\} is monochromatic in every finite coloring of the positive integers. It gives no quantitative bounds and no density version, and the infinite version is false (Hindman 1980).

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof cites the Green-Tao-Ziegler inverse theorem, Tao-Ziegler concatenation, quantitative polynomial equidistribution and nilpotent recurrence as external inputs, adapting Alweiss's rational argument to the integers.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Hindman's conjecture; it gives, for every rr-coloring of N\mathbb N and every mm, distinct a1<⋯<ama_1<\dots<a_m with all nonempty subset sums and products of one color, with extra growth separation. The proof (about 260 KB of TeX, heavy use of higher-order Fourier analysis) was not refereed. No Lean formalization of this manuscript is in the release (no lean/docs/164.md at the pinned commit). The manuscript says the separation does not give an infinite simultaneous sequence, which Hindman's 1980 counterexample rules out anyway.

Source

Changelog1 change

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