Canonical theorems for colored integers with respect to some linear combinations

TL;DR

This paper extends Hindman's theorem to multiple linear combinations with coefficients 1 and -1, establishing canonical results for solutions to infinite systems of equations in colored integers.

Contribution

It generalizes Taylor's canonization of Hindman's theorem from sums to multiple linear combinations with coefficients 1 and -1.

Findings

Extended Hindman's theorem to linear combinations with coefficients 1 and -1.

Established canonical theorems for solutions to infinite systems.

Provided new combinatorial results for colored integers and linear equations.

Abstract

Hindman proved in 1979 that no matter how natural numbers are colored in r colors, for a fixed positive integer r, there is an infinite subset X of numbers and a color t such that for any finite non-empty subset X' of X, the color of the sum of elements from X' is t. Later, Taylor extended this result to colorings with unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman's theorem and parallels the Canonical Ramsey Theorem of Erdos and Rado. We extend Taylor's result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and -1. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.