Question 20
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 if we require the extra condition that, in each class $C$ of the partition, the set of differences
$$
\{ \min \{ n : n \in C \text{ and } n \gt m \} - m : m \in C \}
$$
be bounded?
What if we require the extra condition that, in each class $C$ of the partition, the set of differences
$$
\{ \min \{ n : n \in C \text{ and } n \gt m \} - m : m \in C \}
$$
be bounded?