Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$.
Let $B=\mathbb Z_{\ge 0}-A$.
Define $\ As= \{a+b : a \in A,\ b \in A\}$.
Define $\ Bs=\{a+b : a \in B,\ b \in B\}$.
Then the set $\ \{n,\ n+1,\ n+2, \ ...\}\ \subseteq\ As\ $or $\ Bs$ for some nonnegative integer $n$?