## Abstract

A 1-avoiding set is a subset of \(\mathbb {R}^n\) that does not contain pairs of points at distance 1. Let \(m_1(\mathbb {R}^n)\) denote the maximum fraction of \(\mathbb {R}^n\) that can be covered by a measurable 1-avoiding set. We prove two results. First, we show that any 1-avoiding set in \(\mathbb {R}^n (n\ge 2)\) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other) has density strictly less than \(1/2^n\). For the special case of sets with block structure this proves a conjecture of Erdős asserting that \(m_1(\mathbb {R}^2) < 1/4\). Second, we use linear programming and harmonic analysis to show that \(m_1(\mathbb {R}^2) \le 0.258795\).

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## Notes

This is related to the following observation: Let

*G*be a subgraph of a finite vertex-transitive graph*H*. Then \(\alpha (H) / |V(H)| \le \alpha (G) / |V(G)|\).

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## Acknowledgments

The authors are grateful to the referees whose valuable suggestions helped to improve the presentation of the paper. Part of this research was done when T. Keleti was a visitor at the Alfréd Rényi Institute of Mathematics; he was also supported by OTKA Grant No. 104178. M. Matolcsi and I.Z. Ruzsa were supported by OTKA No. 109789 and ERC-AdG 321104. F.M. de Oliveira Filho was partially supported by FAPESP project 13/03447-6.

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Keleti, T., Matolcsi, M., de Oliveira Filho, F.M. *et al.* Better Bounds for Planar Sets Avoiding Unit Distances.
*Discrete Comput Geom* **55**, 642–661 (2016). https://doi.org/10.1007/s00454-015-9751-5

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DOI: https://doi.org/10.1007/s00454-015-9751-5