Result 164, Combinatorics

Hindman’s finite sums and products conjecture

Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.

Proof

The bigger picture

Why it matters

Addition and multiplication usually impose different demands on a set of numbers. This result claims that any division of the positive integers into finitely many categories leaves arbitrarily large finite sets exhibiting both patterns within one category.

What changes?

The manuscript reports a proof of Hindman's finite sums and products conjecture. A finite coloring assigns each positive integer one of finitely many labels. The claim is that, for every such coloring and every positive integer k, there are k distinct positive integers such that adding or multiplying any nonempty selection of them gives the same color. Each selected number is used once, and one common color covers both sums and products.

What does that help mathematicians do?

For two chosen numbers, the conclusion requires the numbers themselves, their sum and their product to share a color. For larger sets, it controls every nonempty subset simultaneously, not just pairs. Thus a researcher could rule out any finite coloring designed to prevent these joint patterns at a prescribed finite size. The statement does not assert an infinite set with the same property.

Are there practical applications?

The immediate value is foundational: it describes an unavoidable interaction between addition and multiplication under finite partitions of the positive integers. It would supply an existence guarantee for mathematical arguments requiring both kinds of monochromatic structure on the same set. The supplied abstract gives no procedure or numerical search bounds for finding that set.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Monochromatic finite sums and products in the positive integers

September 23, 2026 70 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,
  author = {{OpenAI}},
  title = {{Monochromatic finite sums and products in the positive integers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf}{OAI:Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.