Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Max-norm Ramsey Theory   Given a metric space $\mathcal{M}$ that contains at least two points, the
chromatic number $\chi\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ 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 $\chi\left(\mathbb{R}^n_{\infty}, \mathcal{M}
\right)$ grows exponentially for all finite $\mathcal{M}$. In the present paper
we refine this result by giving the exact value $\chi_{\mathcal{M}}$ such that
$\chi\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right) =
(\chi_{\mathcal{M}}+o(1))^n$ 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 $\chi\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$
tends to infinity as $n \rightarrow \infty$.

35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an