Bounding the size of an almost-equidistant set in Euclidean space

A set of points in d-dimensional Euclidean space is almost equidistant if among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in Rd has cardinality O(d4/3)

On the chromatic numbers of 3-dimensional slices

We prove that for an arbitrary $\varepsilon > 0$ holds $$\chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10,$$ where $\chi(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of monochromatic points at the distance 1 apart

Constructing 5-chromatic unit distance graphs embedded in the euclidean plane and two-dimensional spheres

This paper is devoted to algorithms for finding unit-distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson-Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, this graphs does not contain the Moser spindle as a subgraph. The constructions of 5-chromatic graphs embedded in a sphere at two values of the radius are given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed.Comment: 21 pages, 12 figure