### Abstract

We survey some problems and results around one of Paul Erdős’s favorite questions, first published 70 years ago: What is the maximum number of times that the unit distance can occur among n points in the plane? This simple and beautiful question has generated a lot of important research in discrete geometry, in extremal combinatorics, in additive number theory, in Fourier analysis, in algebra, and in other fields, but we still do not seem to be close to a satisfactory answer.

Original language | English |
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Title of host publication | Open Problems in Mathematics |

Publisher | Springer International Publishing |

Pages | 459-477 |

Number of pages | 19 |

ISBN (Electronic) | 9783319321622 |

ISBN (Print) | 9783319321608 |

DOIs | |

Publication status | Published - Jan 1 2016 |

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### Keywords

- Combinatorial geometry
- Diameter graph
- Geometric graph
- Unit circle
- Unit distance

### ASJC Scopus subject areas

- Mathematics(all)
- Economics, Econometrics and Finance(all)
- Business, Management and Accounting(all)

### Cite this

*Open Problems in Mathematics*(pp. 459-477). Springer International Publishing. https://doi.org/10.1007/978-3-319-32162-2_15

**Erdős’s unit distance problem.** / Szemerédi, E.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Open Problems in Mathematics.*Springer International Publishing, pp. 459-477. https://doi.org/10.1007/978-3-319-32162-2_15

}

TY - CHAP

T1 - Erdős’s unit distance problem

AU - Szemerédi, E.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We survey some problems and results around one of Paul Erdős’s favorite questions, first published 70 years ago: What is the maximum number of times that the unit distance can occur among n points in the plane? This simple and beautiful question has generated a lot of important research in discrete geometry, in extremal combinatorics, in additive number theory, in Fourier analysis, in algebra, and in other fields, but we still do not seem to be close to a satisfactory answer.

AB - We survey some problems and results around one of Paul Erdős’s favorite questions, first published 70 years ago: What is the maximum number of times that the unit distance can occur among n points in the plane? This simple and beautiful question has generated a lot of important research in discrete geometry, in extremal combinatorics, in additive number theory, in Fourier analysis, in algebra, and in other fields, but we still do not seem to be close to a satisfactory answer.

KW - Combinatorial geometry

KW - Diameter graph

KW - Geometric graph

KW - Unit circle

KW - Unit distance

UR - http://www.scopus.com/inward/record.url?scp=85053748549&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053748549&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-32162-2_15

DO - 10.1007/978-3-319-32162-2_15

M3 - Chapter

SN - 9783319321608

SP - 459

EP - 477

BT - Open Problems in Mathematics

PB - Springer International Publishing

ER -