Hindman’s finite sums and products conjecture

However whole numbers are colored, some set of each size has its sums and products in one color, the paper claims.

1 paper September 23, 2026 Explained by AI

Color every positive whole number with one of a few colors. Hindman conjectured that, for any size k, some k numbers have all their subset sums and subset products in a single color. This paper claims to prove that conjecture. The sets are finite; Hindman had shown that infinite versions can fail. No Lean formalization is listed.

First step

1979

Neil Hindman, who posed the finite conjecture

Significance

Major

If the proof holds, it settles a decades-old conjecture that mixes addition and multiplication in Ramsey theory.

Trust

Claimed, not checked

Field

Combinatorics

1 paper

The explanation

Color each positive whole number red or blue by one rule. Write it in binary. If it has an even number of binary digits, color it red; if odd, blue. So 3, written 11, is red. So is 12, written 1100.

Now look for a set whose sums and products all share one color. Take {3, 11}. The sums are 3, 11 and 14. The products are 3, 11 and 33. All four numbers have an even number of binary digits, so all are red. Try three numbers: {3, 12, 228}. Their seven subset sums and seven subset products give 11 different numbers. Every one of them is red.

Hindman's conjecture says this always worksSource 1. Pick any number of colors, any coloring and any size k. Then some k-element set has all its nonempty subset sums and subset products in one colorSource 1. A subset sum adds some of the numbers, each used at most once. A subset product multiplies them the same way.

Sums alone and products alone were handled long ago. Schur's theorem, from 1916, finds x, y and x + y in one colorSource 2. The Folkman–Rado–Sanders theorem extends this to all subset sums of large finite setsSource 3. Recoloring each n by the color of 2ⁿ turns sum results into product resultsSource 4. Mixing both is different. In 1980 Hindman built a coloring with no infinite set whose terms, pairwise sums and pairwise products share a colorSource 5. His finite conjecture asks only for large finite setsSource 6.

Partial results came in steps. Moreira proved in 2017 that x, x + y and xy can share a color, for any finite coloringSource 7. Over the rational numbers, which include fractions, Bowen and Sabok found x, y, x + y and xy in one colorSource 8. Alweiss then proved the full sums-and-products theorem over the rationalsSource 9.

This paper claims to prove the conjecture for the positive integersSource 10. It also claims the numbers can be spread so far apart that no two expressions collide, apart from the numbers themselvesSource 11. Then a k-element set gives 2(2ᵏ − 1) − k different same-colored numbersSource 12. For k = 3 that is 11, as in the example.

The obstacle is moving from fractions to whole numbers. Clearing denominators means multiplying everything by one common number. That scales a sum by one power of it, but a product of j entries by its j-th powerSource 13. A coloring need not respect those different changesSource 13. The new work keeps control of the additive shifts while protecting the product colorsSource 14. It does this by building bounded predictions for where each color appears, then picking scales that keep those predictions positive at the needed sums.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not give an infinite set with this property; Hindman showed in 1980 that such sets need not exist.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major only if the proof holds. It gives finite sets of every size, not infinite ones. No Lean formalization is listed and no specialist review is published.

Words used

Coloring: A rule that gives each positive integer one of finitely many labels, called colors.
Monochromatic: All in one color.
Subset sum: The sum of a nonempty selection of the set's numbers, each used at most once.
Subset product: The product of a nonempty selection of the set's numbers, each used at most once.
Ramsey theory: The study of patterns that must appear in one color, however a large structure is colored.
Rational numbers: Numbers that can be written as fractions of whole numbers.

The original paper

Monochromatic finite sums and products in the positive integers (PDF, opens in a new tab)
September 23, 2026

All quoted lines (14)

We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer $k$ , there is a $k$ -element set whose nonempty subset sums and nonempty subset products all have the same color.
Monochromatic finite sums and products in the positive integers Back to text
Schur's theorem guarantees a monochromatic triple $\{x,y,x+y\}$ in every finite coloring of $\N$ .
Monochromatic finite sums and products in the positive integers Back to text
The finite sums theorem of Folkman–Rado–Sanders extends this to the nonempty subset sums of arbitrarily large finite sets; Sanders's original argument appears in.
Monochromatic finite sums and products in the positive integers Back to text
Applying an additive theorem to the coloring $n\mapsto\chi(2^n)$ gives its multiplicative counterpart.
Monochromatic finite sums and products in the positive integers Back to text
The simultaneous question is different: Hindman constructed a finite coloring of $\N$ with no infinite set whose terms, pairwise sums and pairwise products all have one color.
Monochromatic finite sums and products in the positive integers Back to text
His finite conjecture asks for arbitrarily large finite sets instead; see also.
Monochromatic finite sums and products in the positive integers Back to text
For arbitrary finite colorings, Moreira proved the existence of monochromatic triples $\{x,x+y,xy\}$ as part of a wider theorem on prefix products and polynomial shifts.
Monochromatic finite sums and products in the positive integers Back to text
Bowen and Sabok established $\{x,y,x+y,xy\}$ over $\Q$ , and Alweiss proved the full finite sums and products theorem over $\Q$ .
Monochromatic finite sums and products in the positive integers Back to text
Bowen and Sabok established $\{x,y,x+y,xy\}$ over $\Q$ , and Alweiss proved the full finite sums and products theorem over $\Q$ .
Monochromatic finite sums and products in the positive integers Back to text
Theorem … resolves Hindman's finite sums and products conjecture positively.
Monochromatic finite sums and products in the positive integers Back to text
The separation in (…) also eliminates every collision between expressions except the shared singleton values.
Monochromatic finite sums and products in the positive integers Back to text
In particular, the union has $2(2^m-1)-m$ distinct elements.
Monochromatic finite sums and products in the positive integers Back to text
A common dilation scales a sum by one power of the dilation and a product of $j$ entries by its $j$ th power; an arbitrary coloring need not respect these different changes.
Monochromatic finite sums and products in the positive integers Back to text
The additional task is to control the additive shifts arithmetically while preserving all the required multiplicative colors.
Monochromatic finite sums and products in the positive integers Back to text

Other sources

I. Schur, “Über die Kongruenz xᵐ+yᵐ≡zᵐ (mod p)”, Jahresbericht der DMV 25 (1916), 114–117 (no online copy found)
J. H. Sanders, “A Generalization of Schur's Theorem”, Ph.D. dissertation excerpt, Yale University, 1968; arXiv:1712.03620
N. Hindman, “Partitions and sums and products of integers”, Transactions of the AMS 247 (1979), 227–245
N. Hindman, “Partitions and sums and products—Two counterexamples”, Journal of Combinatorial Theory Series A 29(1) (1980), 113–120
J. Moreira, “Monochromatic sums and products in ℕ”, Annals of Mathematics 185(3) (2017), 1069–1090
M. Bowen and M. Sabok, “Monochromatic products and sums in the rationals”, Forum of Mathematics, Pi 12 (2024), e17
R. Alweiss, “Monochromatic sums and products over ℚ”, arXiv:2307.08901
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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