A. M. Raigorodskii

Coloring Distance Graphs and Graphs of Diameters

In this chapter, we discuss two classical problems lying on the edge of graph theory and combinatorial geometry. The first problem is due to E. Nelson. It consists of coloring metric spaces in such a way that pairs of points at some prescribed distances receive different colors. The second problem is attributed to K. Borsuk and involves finding the minimum number of parts of smaller diameter into which an arbitrary bounded nonsingleton point set in a metric space can be partitioned. Both problems are easily translated into the language of graph theory, provided we consider, instead of the whole space, any (finite) distance graph or any (finite) graph of diameters. During the last decades, a huge number of ideas have been proposed for solving both problems, and many results in both directions have been obtained. In the survey below, we try to give an entire picture of this beautiful area of geometric combinatorics.

On the chromatic numbers of spheres in ℝ n

AbstractLet χ(Srn−1)) be the minimum number of colours needed to colour the points of a sphere Srn−1 of radius

Coloring some finite sets in {R}^{n}

r \geqslant \tfrac{1} {2}

New estimates in the problem of the number of edges in a hypergraph with forbidden intersections

in ℝn so that any two points at the distance 1 apart receive different colours. In 1981 P. Erdős conjectured that χ(Srn−1)→∞ for all

Combinatorial Geometry and Coding Theory

r \geqslant \tfrac{1} {2}

Distance graphs with large chromatic number and without large cliques

. This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(Srn−1) ≥ n. In the same paper, Lovász claimed that if