In this paper, we obtain a lower bound on the number of edges in a unit distance graph Γ in an infinitesimal plane layer ℝ2 × [0, ε]d, which relates the number of edges e(Γ), the number of vertices ν(Γ), and the independence number α(Γ). Our bound \( e\left(\varGamma \right)\ge \frac{19\nu \left(\varGamma \right)-50\alpha \left(\varGamma \right)}{3} \) is a generalization of a previous bound for distance graphs in the plane and a strong improvement of Turán’s bound in the case where \( \frac{1}{5}\le \frac{\alpha \left(\varGamma \right)}{v\left(\varGamma \right)}\le \frac{2}{7} \).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 464, 2017, pp. 132–168.
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Shabanov, L.E. Turán-Type Results for Distance Graphs in an Infinitesimal Plane Layer. J Math Sci 236, 554–578 (2019). https://doi.org/10.1007/s10958-018-4133-1
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DOI: https://doi.org/10.1007/s10958-018-4133-1