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Hindman's finite sums and products conjecture

Theorem 1.1: for every finite coloring of $\mathbb N$ and every $m\ge1$ there are $a_1<\dots<a_m$ such that every nonempty subset sum and subset product has one common color; the $a_d$ can be made to grow faster than any prescribed power of the earlier sums and products, so all $2(2^m-1)-m$ expressions are distinct. In particular $\{x,y,x+y,xy\}$ 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).

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Hindman's finite sums and products conjecture

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Unreleased internal OpenAI model
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VibeMathed record: Hindman's finite sums and products conjecture

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Hindman's conjecture; it gives, for every $r$ -coloring of $\mathbb N$ and every $m$ , distinct $a_1<\dots<a_m$ 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.

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Hindman's finite sums and products conjecture

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