Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15):
It has been shown that there is a large $M$ so that it is possible to partition $\mathbb{E}^2$ into
two sets $A$ and $B$ so that $A$ contains no pair of points with distance 1 and $B$ contains no A.P.
of length $M$.
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