Erdős Problem #1090

Let \(k\geq 3\). Does there exist a finite set \(A\subset \mathbb{R}^2\) such that, in any \(2\)-colouring of \(A\), there exists a line which contains at least \(k\) points from \(A\), and all the points of \(A\) on the line have the same colour? Erdős [Er75f] says Graham and Selfridge proved the answer is yes when \(k=3\). Hunter has observed that, for sufficiently large \(n\), a generic projection of \([k]^n\) into \(\mathbb{R}^2\) has this property, by the Hales-Jewett theorem.