Chromatic number of spacetime
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912121189564416 |
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| author | Davies, James |
| author_facet | Davies, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16885 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chromatic number of spacetime Davies, James Combinatorics Mathematical Physics Metric Geometry Number Theory 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. |
| title | Chromatic number of spacetime |
| topic | Combinatorics Mathematical Physics Metric Geometry Number Theory |
| url | https://arxiv.org/abs/2308.16885 |