MathDB

C3

Part of 2011 Balkan MO Shortlist

Problems(1)

partition the set of positive integers into two class

Source: Balkan MO ShortList 2011 C3

4/6/2020
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 C\mathcal{C} 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?
combinatoricspartition