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,‖x−y‖2≠1. In this paper we study the number
m1(R2) which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, m1(R2) 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 χf(R2) of the plane.
We establish that m1(R2)≤0.25647 and χf(R2)≥3.8991.