Geodesic
Borsuk's problem and the chromatic numbers of some metric spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 56 (2001) no. 1, pp. 103-139

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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. 
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     author = {A. M. Raigorodskii},
     title = {Borsuk's problem and the chromatic numbers of some metric spaces},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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A. M. Raigorodskii. Borsuk's problem and the chromatic numbers of some metric spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 56 (2001) no. 1, pp. 103-139. http://geodesic.mathdoc.fr/item/RM_2001_56_1_a2/