All finite sets are Ramsey in the maximum norm
Forum of Mathematics, Sigma, Tome 9 (2021)
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For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$. In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$.
@article{10_1017_fms_2021_50,
author = {Andrey Kupavskii and Arsenii Sagdeev},
title = {All finite sets are {Ramsey} in the maximum norm},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {9},
year = {2021},
doi = {10.1017/fms.2021.50},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.50/}
}
Andrey Kupavskii; Arsenii Sagdeev. All finite sets are Ramsey in the maximum norm. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.50
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