TITLE: Is it possible to partition $\mathbb{N}_+$ into a *finite* family of sets completely not closed under $+$?
QUESTION [6 upvotes]: Let's say that $A \subseteq \mathbb{N}_+$ is completely not closed under $+$ if
$$
\forall_{a,b \in A}[{a+b \notin A}]
$$
Is it possible to partition $\mathbb{N}_+$ into a finite family of sets completely not closed under $+$?

REPLY [7 votes]: No. This is because of the following Ramsey-theoretic principle proved by Schur: for any $k$, there exists $S(k)$ such that if $\{ 1, 2, ... S(k) \}$ (this might be off by $1$) is partitioned into $k$ disjoint subsets $A_1, ..., A_k$, then some $A_i$ has the property that there exists $a, b, c \in A_i$ such that $a + b = c$.