Borsuk's problem and the chromatic numbers of some metric spaces

A detailed survey is given of various results pertaining to two well-known problems of combinatorial geometry: Borsuk's problem on partitions of an arbitrary bounded $d$-dimensional set of non-zero diameter into parts of smaller diameter, and the problem of finding chromatic numbers of some metric spaces. Furthermore, a general method is described for obtaining good lower bounds for the minimum number of parts of smaller diameter into which an arbitrary non-singleton set of dimension $d$ can be divided as well as for the chromatic numbers of various metric spaces, in particular, $\mathbb R^d$ and $\mathbb Q^d$. Finally, some new lower bounds are proved for chromatic numbers in low dimensions, and new natural generalizations of the notion of chromatic number are proposed.