Max-norm Ramsey Theory
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914677983805440 |
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| author | Frankl, Nóra Kupavskii, Andrey Sagdeev, Arsenii |
| author_facet | Frankl, Nóra Kupavskii, Andrey Sagdeev, Arsenii |
| contents | Given a metric space $\mathcal{M}$ that contains at least two points, the chromatic number $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ is defined as the minimum number of colours needed to colour all points of an $n$-dimensional space $\mathbb{R}^n_{\infty}$ with the max-norm such that no isometric copy of $\mathcal{M}$ is monochromatic. The last two authors have recently shown that the value $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ grows exponentially for all finite $\mathcal{M}$. In the present paper we refine this result by giving the exact value $χ_{\mathcal{M}}$ such that $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right) = (χ_{\mathcal{M}}+o(1))^n$ for all 'one-dimensional' $\mathcal{M}$ and for some of their Cartesian products. We also study this question for infinite $\mathcal{M}$. In particular, we construct an infinite $\mathcal{M}$ such that the chromatic number $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ tends to infinity as $n \rightarrow \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_08949 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Max-norm Ramsey Theory Frankl, Nóra Kupavskii, Andrey Sagdeev, Arsenii Combinatorics 05D10, 05C15, 52C10 Given a metric space $\mathcal{M}$ that contains at least two points, the chromatic number $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ is defined as the minimum number of colours needed to colour all points of an $n$-dimensional space $\mathbb{R}^n_{\infty}$ with the max-norm such that no isometric copy of $\mathcal{M}$ is monochromatic. The last two authors have recently shown that the value $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ grows exponentially for all finite $\mathcal{M}$. In the present paper we refine this result by giving the exact value $χ_{\mathcal{M}}$ such that $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right) = (χ_{\mathcal{M}}+o(1))^n$ for all 'one-dimensional' $\mathcal{M}$ and for some of their Cartesian products. We also study this question for infinite $\mathcal{M}$. In particular, we construct an infinite $\mathcal{M}$ such that the chromatic number $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ tends to infinity as $n \rightarrow \infty$. |
| title | Max-norm Ramsey Theory |
| topic | Combinatorics 05D10, 05C15, 52C10 |
| url | https://arxiv.org/abs/2111.08949 |