  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.
