The density of sets avoiding distance 1 in Euclidean space

Christine Bachoc (1) , Alberto Passuello (2) , Alain Thiery (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Équipe Théorie des Nombres

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.

Mots clés: unit distance graph; measurable chromatic number; theta number; linear programming

Domaines: Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

Christine Bachoc, Alberto Passuello, Alain Thiery. The density of sets avoiding distance 1 in Euclidean space. 2015. ⟨hal-00935665v2⟩