О рациональных аналогах проблем Нелсона - Хадвигера

В этой статье мы рассматриваем аффинно-рациональные аналоги задачи Нелсона-Хадвигера о нахождении хроматического числа рационального пространства и задачи Борсука о разбиении на части меньшего диаметра. Доказаны новые нижние оценки, в частности улучшены оценки минимального контрпримера для гипотезы Борсука

Counterexamples to Borsuk’s Conjecture with Large Girth

Borsuk’s celebrated conjecture, which has been disproved, can be stated as follows: in ℝn, there exist no diameter graphs with chromatic number larger than n + 1. In this paper, we prove the existence of counterexamples to Borsuk’s conjecture which, in addition, have large girth. This study is in the spirit of O’Donnell and Kupavskii, who studied the existence of distance graphs with large girth. We consider both cases of strict and nonstrict diameter graphs. We also prove the existence of counterexamples with large girth to a statement of Lovász concerning distance graphs on the sphere

Chromatic numbers of spheres

The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds on chromatic numbers of spheres. This also allows us to give new upper bounds on chromatic numbers of any bounded subsets

Space programs summary no. 37-66, volume 3 for the period 1 October - 30 November 1970. Supporting research and advanced development

Research studies on development of Thermoelectric Outer Planet Spacecraft /TOPS/ and lunar exploratio