On the density of sets of the Euclidean plane avoiding distance 1
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A subset A ⊂ R^2 is said to avoid distance 1 if: ∀ x,y ∈ A, x-y _2 ≠ 1. In this paper we study the number m_1( R^2) which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, m_1( 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 χ_f( R^2) of the plane. We establish that m_1( R^2) ≤ 0.25646 and χ_f( R^2) ≥ 3.8992.
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