Thomas Bellitto ; Arnaud Pêcher ; Antoine Sédillot - On the density of sets of the Euclidean plane avoiding distance 1

dmtcs:5153 - Discrete Mathematics & Theoretical Computer Science, August 31, 2021, vol. 23 no. 1 - https://doi.org/10.46298/dmtcs.5153
On the density of sets of the Euclidean plane avoiding distance 1Article

Authors: Thomas Bellitto ; Arnaud Pêcher ; Antoine Sédillot

    A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ 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 $\chi_f(\mathbb R^2) \geq 3.8991$.

    Comment: 13 pages, 8 figures

    https://doi.org/10.46298/dmtcs.5153
    Source: arXiv.org:1810.00960
    Volume: vol. 23 no. 1
    Section: Combinatorics
    Published on: August 31, 2021
    Accepted on: March 22, 2021
    Submitted on: January 31, 2019
    Keywords: Mathematics - Metric Geometry, Computer Science - Discrete Mathematics, Mathematics - Combinatorics
    Licence: arXiv.org - Non-exclusive license to distribute
    Funding:
      Source : OpenAIRE Graph
    • Cuts and decompositions: algorithms and combinatorial properties; Funder: European Commission; Code: 714704

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