partition the set of positive integers into two class
Source: Balkan MO ShortList 2011 C3
April 6, 2020
combinatoricspartition
Problem Statement
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 of the partition, the set of difference
\begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*}
be bounded?