Covering sets in $\mathbb{R}^m$

The paper investigates questions related to Borsuk's classical problem of partitioning a set in Euclidean space into subsets of smaller diameter, as well as to the well-known Nelson-Erdős-Hadwiger problem on the chromatic number of a Euclidean space.
The results of the work are obtained using combinatorial and geometric methods alike. A new approach to the investigation of such problems is suggested; it leads to a collection of results which significantly improve all results known so far.
Bibliography: 58 titles.