2023/07/18 by Ryan Alweiss, Alweiss, Ryan
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2307.08901
openalex publication_date 2023/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hindman's finite sums theorem states that in any finite coloring of the naturals, there is an infinite sequence so that all of its finite subset sums are the same color. In 1979, Hindman showed that there is a finite coloring of the naturals so that no infinite sequence has all of its pairwise sums and pairwise products the same color. Hindman conjectured that for any n, a finite coloring of the naturals contains n numbers all of whose subset sums and subset products are the same color. In this paper we prove the version of this statement where we color the rationals instead of the integers. In other words, we show that the pattern \ ∑i ∈ Sxi, ∏i ∈ Sxi \, where S ranges over all nonempty subsets of [n], is partition regular over the rationals.