On the measurable chromatic number of a space of dimension \(n \leq 24\). (English. Russian original) Zbl 1334.05034
Dokl. Math. 92, No. 3, 761-763 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 465, No. 6, 647-650 (2015).
Summary: This paper is devoted to the classical problem of finding the measurable chromatic number of an \(n\)-dimensional Euclidean space, i.e., the value \(\chi_m(\mathbb{R}^n)\) equal to the least possible number of Lebesgue measurable sets that do not contain pairs of points at a distance of 1 and cover the whole space. Assuming that a certain hypothesis is true, we significantly improve the lower bounds for \(\chi_m(\mathbb{R}^n)\).
MSC:
| 05C15 | Coloring of graphs and hypergraphs |
| 05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |
Keywords:
Lebesgue measurable setsSoftware:
CVXOPTReferences:
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