Author:
Kupavskii Andrey,Sagdeev Arsenii
Abstract
Abstract
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
$\mathcal M$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
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