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.

Manuscript

Monochromatic finite sums and products in the positive integers

September 23, 2026 · source · cite as \cite{OAI:Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026}

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.

Result 164 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.