partitioning the integers without arithmetic progressions What is the least $n$ such that the set of all integers can be partitioned into $n$ disjoint subsets, none of which contain any infinite arithmetic progressions (arbitrarily long but finite arithmetic progressions are allowed)? In particular, can it be done for $n=2$? I tried e.g. splitting the integers into perfect powers and non-powers but then $4n+2$ is never a perfect power. 

I believe something like $X_i=[-10^i,-10^{i-1})$ and $Y_i=[10^{i-1},10^i)$, and then take $$A\;=\;\dots,X_7,X_5,X_3,X_1,0,Y_1,Y_3,Y_5,\dots$$ and $$B\;=\;\dots, X_6,X_4,X_2,-1,Y_2,Y_4,Y_6,\dots$$ to be the 2 parts partitioning the integers. As the gaps in either part increase in both directions, any arithmetic progression with common difference $d$ must be finite. This is because as you go far enough along $A$ (or $B$) in either direction, you will find a gap of length greater than $2d$.
