What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?

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 $M$.