Turán Type Results for Distance Graphs

Abstract

The classical Turan theorem determines the minimum number of edges in a graph on n vertices with independence number $$\alpha $$ź. We consider unit-distance graphs on the Euclidean plane, i.e., graphs $$ G = (V,E) $$G=(V,E) with $$ V \subset {\mathbb {R}}^2 $$VźR2 and $$ E = \{\{\mathbf{x}, \mathbf{y}\}: |\mathbf{x}-\mathbf{y}| = 1\} $$E={{x,y}:|x-y|=1}, and show that the minimum number of edges in a unit-distance graph on n vertices with independence number $$ \alpha \leqslant \lambda n $$źźźn, $$ \lambda \in [\frac{1}{4}, \frac{2}{7}] $$źź[14,27], is bounded from below by the quantity $$ \frac{19 - 50 \lambda }{3} n $$19-50ź3n, which is several times larger than the general Turan bound and is tight at least for $$ \lambda = \frac{2}{7} $$ź=27.