This article is cited in
12 papers
Covering planar sets
V. P. Filimonov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
Problems connected with the classical Borsuk problem on partitioning a set in Euclidean space into subsets of smaller diameter, and also connected with the Nelson-Hadwiger problem on the chromatic number of Euclidean space, are studied. New bounds are obtained for the quantities
$d_n=\sup d_n(\Phi)$ and
$d'_n=\sup d'_n(\Phi)$, where the suprema are taken over all sets of unit diameter on a plane, and where the quantities
$d_n(\Phi)$ and
$d'_n(\Phi)$ are defined for a given bounded set
$\Phi\subset\mathbb{R}^2$ as follows:
\begin{align*}
d_n(\Phi)&=\inf\{x\in\mathbb{R}^+:\Phi\subseteq
\Phi_1\cup\dots\cup\Phi_n,\,\forall\, i\ \operatorname{diam}\Phi_i\le x\},
\\
d'_n(\Phi)&=\inf\{x\in\mathbb{R}^+:\Phi\subseteq
\Phi_1\cup\dots\cup\Phi_n,\,\forall\, i\ \forall\, X,Y\in\Phi_i\,\ XY\ne x\}.
\end{align*}
Here the
$\Phi_i\subset\mathbb R^2$ are subsets,
$\operatorname{diam}\Phi_i$ is the diameter of
$\Phi_i$,
$XY$ is the distance between the points
$X$ and
$Y$, and
$n\in \mathbb N$. The bounds obtained for
$d_n$ are better than any known before; this paper is the first to consider the values
$d'_n$.
Bibliography: 19 titles.
Keywords:
chromatic number, Borsuk problem, diameter of a set, coverings of planar sets, universal covering sets and systems.
UDC:
514.174
MSC: 52C15 Received: 27.05.2008 and 24.08.2009
DOI:
10.4213/sm6369
Fulltext:
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