partition the set of positive integers into two classes

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Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class 42ecbfb5e7dd547c1e0ca2eeb4669f2cc7f1ae4e of the partition, the set of difference
7782e7fbad91e6a8671ac339b067fd722e361800be bounded?

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