The density of sets avoiding distance 1 in Euclidean space
| Autor: | Alberto Passuello, Alain Thiery, Christine Bachoc |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Équipe Théorie des Nombres, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), CPU |
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
measurable chromatic number
Magnitude (mathematics) 0102 computer and information sciences Euclidean distance matrix 01 natural sciences Upper and lower bounds Theoretical Computer Science Combinatorics Mathematics - Metric Geometry 52C10 90C05 90C27 05C69 FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Unit distance graph unit distance graph 0101 mathematics [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] Mathematics Discrete mathematics Euclidean space 010102 general mathematics Metric Geometry (math.MG) linear programming [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Exponential function Euclidean distance Computational Theory and Mathematics 010201 computation theory & mathematics Distance from a point to a plane Combinatorics (math.CO) Geometry and Topology [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] theta number |
| Popis: | We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovasz theta number and of a combinatorial argument involving finite subgraphs of the unit distance graph. In turn, we straightforwardly obtain an asymptotic improvement for the measurable chromatic number of Euclidean space. We also tighten previous results for the dimensions between 4 and 24. Revised version, to appear in Discrete and Computational Geometry |
| Databáze: | OpenAIRE |
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