# Turán Type Results for Distance Graphs

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## Abstract

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

## Keywords

Turán theorem Independence number Distance graphs## Mathematics Subject Classification

05C35 52C10## Notes

### Acknowledgments

This work is done under the financial support of the following grants: the Grant 15-01-00350 of Russian Foundation for Basic Research, the Grant NSh-2964.2014.1 supporting leading scientific schools of Russia.

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