Turán-type bounds for distance graphs

Abstract

A lower bound is obtained for the number of edges in a distance graph G in an infinitesimal plane layer ℝ² × [0, ε]^d, which relates the number of edges e(G), the number of vertices ν(G), and the independence number α(G). It is proved that $e\left( G \right) \geqslant \frac{{19\nu \left( G \right) - 50\alpha \left( G \right)}}{3}$. This result generalizes a previous bound for distance graphs in the plane. It substantially improves Turán’s bound in the case where $\frac{1}{5} \leqslant \frac{{\alpha \left( G \right)}}{{\nu \left( G \right)}} \leqslant \frac{2}{7}$.