Canonical Folkman Theorem

Statement

Let $FS(\{x_i\}_{i = 1}^n)$ denote the set of all finite sums of elements of $\{x_i\}_{i = 1}^n$. Let $C$ be a (possibly infinite) coloring of the positive integers, and let $n$ be an arbitrary positive integer. There exists $\{x_i\}_{i = 1}^n$ such that at least one of the following 3 conditions holds.

\1) $FS(\{x_i\}_{i = 1}^n)$ is a monochromatic set.

\2) $FS(\{x_i\}_{i = 1}^n)$ is a rainbow set.

\3) For any $B \subseteq [1,n]$, the color of $\sum_{i \in B}x_i$ is determined solely by $\min(B)$.