Covering planar sets

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 $d'_n$.
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