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

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

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

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

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 $n$?

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