Chromatic number of spacetime

Abstract: We observe that an old theorem of Graham implies that for any positive integer $s$, there exists some positive integer $T(s)$ such that every $s$-colouring of $\mathbb{Z}^2$ contains a monochromatic pair of points $(x,y),(x',y')$ with $(x-x')^2 - (y-y')^2 = (T(s))^2$. By scaling, this implies that every finite colouring of $\mathbb{Q}^2$ contains a monochromatic pair of points $(x,y),(x',y')$ with $(x-x')^2 - (y-y')^2 = 1$, which answers in a strong sense a problem of Kosheleva and Kreinovich on a pseudo-Euclidean analogue of the Hadwiger-Nelson problem. The proof of Graham's theorem relies on repeated applications of van der Waerden's theorem, and so the resulting function $T(s)$ grows extremely quickly. We give an alternative proof in the weaker setting of having a second spacial dimension that results in a significantly improved bound. To be more precise, we prove that for every positive integer $s$ with $r\equiv 2 \pmod{4}$, every $s$-colouring of $\mathbb{Z}^3$ contains a monochromatic pair of points $(x,y,z),(x',y',z')$ such that $(x-x')^2 + (y-y')^2 - (z-z')^2 = (5^{(s-2)/4}(8\cdot 5^{(s-2)/2})!)^2$. In fact, we prove a stronger density version. The density version in $\mathbb{Z}^2$ remains open.