On the density of sets of the Euclidean plane avoiding distance 1Article
Authors: Thomas Bellitto ; Arnaud Pêcher ; Antoine Sédillot
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Thomas Bellitto;Arnaud Pêcher;Antoine Sédillot
A subset A⊂R2 is said to avoid distance 1 if: ∀x,y∈A,‖ In this paper we study the number
m_1(\mathbb R^2) which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, m_1(\mathbb
R^2) represents the highest proportion of the plane that can be filled by a
set avoiding distance 1. This parameter is related to the fractional chromatic
number \chi_f(\mathbb R^2) of the plane.
We establish that m_1(\mathbb R^2) \leq 0.25647 and \chi_f(\mathbb R^2)
\geq 3.8991.