Turán-Type Results for Distance Graphs in an Infinitesimal Plane Layer

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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, e]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 Turan’s bound in the case where $$ \frac{1}{5}\le \frac{\alpha \left(\varGamma \right)}{v\left(\varGamma \right)}\le \frac{2}{7} $$ .