---
tag: K613A63D6N
kind: statement
role: theorem
name: Canonical Folkman Theorem
sources:
  - wikipedia: en:29265641@r1324819265
    locator: Folkman's theorem, section Canonical Folkman Theorem
    note: adopted
    mention: wikipedia://en/29265641#ZM7W
    sections: [Statement]
inventory: [6ND99RERZZ]
---

## 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 $\min(B)$.