Monochromatic Sums and Products over $\mathbb{Q}$

TL;DR

This paper proves a partition regularity result over the rationals, showing that for any finite coloring, one can find n numbers with all subset sums and products sharing the same color, extending Hindman's theorem.

Contribution

It extends Hindman's finite sums theorem to include subset products over the rationals, confirming a conjecture for the rational numbers.

Findings

Partition regularity of sum-product patterns over rac{}

Existence of monochromatic configurations for any finite coloring over rac{}

Generalization of Hindman's theorem to sum-product patterns over rac{}

Abstract

Hindman's finite sums theorem states that in any finite coloring of the naturals, there is an infinite sequence all of whose 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 $\{ \sum_{i \in S}x_i, \prod_{i \in S}x_i \}$, where $S$ ranges over all nonempty subsets of $[n]$, is partition regular over the rationals.