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

Abstract

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 \( \frac{1}{5}\le \frac{\alpha \left(\varGamma \right)}{v\left(\varGamma \right)}\le \frac{2}{7} \).