Andrei Mikhailovich Raigorodskii

Colorings of the space ℝn with several forbidden distances

The paper is concerned with the classical problem concerning the chromatic number of a metric space, i.e., the minimal number of colors required to color all points in the space so that the distance (the value of the metric) between points of the same color does not belong to a given set of positive real numbers (the set of forbidden distances). New bounds for the chromatic number are obtained for the case in which the space is ℝn with a metric generated by some norm (in particular, lp) and the set of forbidden distances either is finite or forms a lacunary sequence.

Chromatic numbers of spaces with forbidden monochromatic triangles

New lower estimates for chromatic numbers of Euclidean spaces with forbidden monochromatic isosceles triangles are obtained.

New lower bound for the chromatic number of a rational space with one and two forbidden distances

A new lower bound for the chromatic number χ(ℚn) of the space ℚn is obtained.

New upper bounds for the independence numbers of graphs with vertices in {−1, 0, 1}n and their applications to problems of the chromatic numbers of distance graphs

Upper bounds for the independence numbers in the graphs with vertices at {−1, 0, 1}n are improved. Their applications to problems of the chromatic numbers of distance graphs are studied.

On the realization of subgraphs of a random graph by diameter graphs in Euclidean spaces

The present paper is motivated by Borsuk’s classical problem of the partition of sets in spaces into parts of smaller diameter. We obtain sharp estimates for the maximal number of vertices of the induced subgraph of a random graph that, with high probability, is isomorphic to the diameter graph with given chromatic number in a space of any fixed dimension.

On a series of problems related to the Borsuk and Nelson-Erdős-Hadwiger problems

In the present paper, a series of problems connecting the Borsuk and Nelson-Erdős-Hadwiger classical problems in combinatorial geometry is considered. The problem has to do with finding the number χ(n, a, d) equal to the minimal number of colors needed to color an arbitrary set of diameter d in n-dimensional Euclidean space in such a way that the distance between points of the same color cannot be equal to a. Some new lower bounds for the quantity χ(n, a, d) are obtained.