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Given a positive integer $n\in\mathbb{N}$ we define the zebra crossing associated with $n$ to be the set $$Z_n = \{[2kn, 2kn+(n-1)] \cap \mathbb{N}: k\in\mathbb{N}\}.$$

Is there an infinite set $A\subseteq\mathbb{N}$ such that $A\cap Z_n$ is finite for all positive integers $n$?

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Take the $k$th element of $A$ to be $a_k:=2k!-1$, for all $k\ge 1$. If $k\ge n$, then $a_k\equiv -1\!\!\pmod{2n}$, whence $a_k\notin Z_n$. Thus, for any fixed $n$, there are only finitely many elements of $A$ contained in $Z_n$.

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  • $\begingroup$ Beat me by 3 seconds! $\endgroup$
    – Wojowu
    Commented Dec 17, 2017 at 13:49
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    $\begingroup$ @Wojowu: If I knew, I would make it 30 seconds! :-) $\endgroup$
    – Seva
    Commented Dec 17, 2017 at 13:50
  • $\begingroup$ @Wojowu: and, your deleted answer beats my by one line! :-) $\endgroup$
    – Seva
    Commented Dec 17, 2017 at 13:52

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