Let k∈N and let a finite coloring of N be given. Generalized Hindman's conjecture. There exist infinitely many x_1<…<x_k∈N such that all finite sums and all finite products formed using distinct elements from x_1,…,x_k are monochromatic. This extends Hindman's sum-and-product problem from two variables to arbitrarily many selected elements; the supplied status evidence says that this conjecture remains open.

Hindman's generalized conjecture on finite sums and products

About 1 year old · traced to 2025

Let $k\in\mathbb{N}$ and let a finite coloring of $\mathbb{N}$ be given. Generalized Hindman's conjecture. There exist infinitely many $x_1<\ldots<x_k\in\mathbb{N}$ such that all finite sums and all finite products formed using distinct elements from $\{x_1,\ldots,x_k\}$ are monochromatic. This extends Hindman's sum-and-product problem from two variables to arbitrarily many selected elements; the supplied status evidence says that this conjecture remains open.

References

Primary source

Florian K. Richter, “Sums and products in sets of positive density”, arXiv:2507.00515 (2026).

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RemarkAI-assistedClaimed by OpenAI. For every finite coloring of the positive integers and every finite k>=1, the manuscript claims k distinct increasing integers whose nonempty subset sums and products all share one color. The smallest integer may exceed any prescribed cutoff, yielding infinitely many such finite tuples as asked on this page. This does not assert one infinite sequence whose sums and products are all monochromatic.See full solutionHide full solution

Claimed by OpenAI. For every finite coloring of the positive integers and every finite k>=1, the manuscript claims k distinct increasing integers whose nonempty subset sums and products all share one color. The smallest integer may exceed any prescribed cutoff, yielding infinitely many such finite tuples as asked on this page. This does not assert one infinite sequence whose sums and products are all monochromatic.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf

OpenAI-164-01-Monochromatic-finite-sums-and-products-in-the-positive-integers.pdf775,466 bytesOpen