Canonical Theorems for Colored Integers with Respect to Some Linear Combinations

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 609-628, March 2024.
Abstract. Hindman proved in 1979 that no matter how natural numbers are colored in [math] colors, for a fixed positive integer [math], there is an infinite subset [math] of numbers and a color [math] such that for any finite nonempty subset [math] of [math], the color of the sum of elements from [math] is [math]. Later, Taylor extended this result to colorings with an 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 Erdős and Rado. We extend Taylor’s result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and [math]. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.