Hindman proved?

The claim, in plain words

Colour the positive integers with finitely many colours. Schur’s theorem gives x, y and x+y in one colour; the finite sums theorem of Folkman, Rado and Sanders gives all subset sums of arbitrarily large finite sets; and “applying an additive theorem to the coloring n ↦ χ(2ⁿ) gives its multiplicative counterpart”. Hindman asked for sums and products at once. He constructed a finite colouring with no infinite set whose terms, pairwise sums and pairwise products all share a colour, and conjectured that finite sets of every size exist. As the paper recounts, Graham and Hindman’s computer-assisted arguments gave {x, y, x+y, xy} in every two-colouring, Bowen proved a two-colour version with distinct, large x and y, Moreira proved {x, x+y, xy} for every finite colouring, and over the rationals Alweiss proved the full finite sums and products theorem. “The passage from rational to integer configurations requires more than clearing denominators”: a dilation scales a sum once and a product of j numbers j times.

Result 164 is a single OpenAI preprint, Monochromatic finite sums and products in the positive integers (70 pages, dated September 23, 2026). Theorem 1.1: for all r, m ≥ 1 and every colouring χ of ℕ with r colours, “there are distinct positive integers a₁ < ⋯ < a_m and a color c ∈ [r] such that χ(∑_j∈J a_j) = χ(∏_j∈J a_j) = c for every ∅ ≠ J ⊆ [m]”, and the numbers can be chosen with a₁ > R and each a_d > R(∑_k<d a_k + ∏_k<d a_k)^D for any fixed R ≥ 2 and D ≥ 1. “Theorem 1.1 resolves Hindman’s finite sums and products conjecture positively.” With this separation the 2^m − 1 sums are distinct, the 2^m − 1 products are distinct, and the two lists meet only at the a_d (Corollary 1.2): for three numbers, eleven values in one colour.

What it doesn’t claim

No infinite set: “the separation does not assert the existence of an infinite simultaneous sequence”, and Hindman showed that one can fail. No explicit set and no bound: “no numerical bound for the smallest configuration is claimed”, and the finite-interval form (Corollary 2.6) gives an N with “no numerical estimate”. The colour is whatever the proof finds. Nothing is claimed for density versions or other number systems; the result over the rationals was already Alweiss’s, and the paper adapts his arguments. As our own illustration, large numbers can be forced: colour n ≥ 2 by the parity of j, where 2^2j ≤ n < 2^2j+1. Two different numbers in one band multiply into the next band, so any three numbers above one whose sums and products all share a colour include one of at least 65,536 (attained by 2, 16, 65,536), and any four, one of at least 2⁶⁴ ≈ 1.8 × 10¹⁹.

Where it stands

Posted as a preprint

OpenAI, 70 pages, dated September 23, 2026

Peer reviewed

Not yet

Checked in Lean

None listed; uncatalogued Lean covers one of the proof’s two principles, not the theorem

Last checked October 19, 2026. The preprint has not been revised since it was added to the repository in its initial commit on October 6, 2026. The release catalog lists no Lean formalization for this result, and none appears in the repository’s formalization list. The repository does contain Lean code, not linked from the catalog, whose top statement (RawMenu.lean) is, by our reading, the paper’s Alignment principle (Principle 2.4), with integer shifts, which makes it slightly stronger: by our count 283 files and 94,076 lines excluding Mathlib (the SumProduct development and supporting nilpotent charts). By our scan it has no unproved steps, and we have not rebuilt it. The Prediction principle, the product step (Lemma 2.5), the final deduction and Theorem 1.1 itself are not formalized; the Lean never mentions a colouring of the integers. The paper is not among those OSVerify reviewed, and we have seen no expert review.

How the proof goes

1. Products first. Each a_d is a product of a block of randomly chosen raw numbers at hugely separated scales. Whatever raw numbers come up, a Ramsey argument and the finite sums theorem pick blocks, depending on them, so that every product of the block products has one colour c (Lemma 2.5, following Alweiss’s argument over the rationals).
2. Sums as nudges. Because the scales are so far apart, a sum of blocks is its largest block plus a comparatively small shift, and an arbitrary colouring can change at every step.
3. Prediction. Inside the relevant counts, higher-order Fourier analysis replaces each colour by a bounded model built from nilsequences, values F(g^kx) read off an orbit in a nilmanifold; where a colour occurs, its model is very likely large. Divisor weights and a transference argument handle the uneven spread of products; the external inputs include the Green–Tao–Ziegler inverse theorem with its correction.
4. Alignment. Scale multipliers from a fixed finite menu make “model above 2τ at the centre” imply “model above τ after every required shift”, with probability at least δ, where δ does not depend on the models’ complexity. It uses nilpotent polynomial recurrence and adapts Alweiss’s compensating scale updates.
5. The order of choices. δ is fixed first, then the tolerances, then the models; the weighted count of good configurations is then positive, so an actual set of integers exists.

The numbers in the video

• Our smallest quartets x, y, x+y, xy in one colour (2 ≤ x < y): parity 2, 4, 6, 8; the parity of the number of prime factors 4, 6, 10, 24 (sums alone already at 2, 3, 5); a random two-colouring 5, 7, 12 for Schur’s triple.
• Our separated triple 3, 13, 111 (each above twice the sum plus product of the earlier ones): 7 sums and 7 products, 11 distinct values.
• Our band colouring: 65,536 for three numbers and 2⁶⁴ ≈ 1.8 × 10¹⁹ for four.
• The Lean count: 283 files and about 94,000 lines, our count of the import closure.

Read the source

• Monochromatic finite sums and products in the positive integers (PDF), 70 pages.
• The uncatalogued Lean statement of the Alignment principle.
• The collection’s catalog, where this is result 164.

The paper

OpenAI groups related papers into one result; this one has a single paper.

• Monochromatic finite sums and products in the positive integers

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.