## Abstract

A set A of nonnegative integers is called a Sidon set if there is no Sidon 4-tuple, i.e., (a, b, c, d) in A with a + b = c + d and {a, b} ∩ {c, d} = ∅. Cameron and Erdős proposed the problem of determining the number of Sidon sets in [n]. Results of Kohayakawa, Lee, Rödl and Samotij, and Saxton and Thomason have established that the number of Sidon sets is between 2^{(1}.16+o(1))√^{n} and 2^{(6}.442+o(1))√^{n}. An α-generalized Sidon set in [n] is a set with at most α Sidon 4-tuples. One way to extend the problem of Cameron and Erdős is to estimate the number of α-generalized Sidon sets in [n]. We show that the number of (n/ log^{4} n)-generalized Sidon sets in [n] with additional restrictions is 2^{Θ(}√n) . In particular, the number of (n/ log^{5} n)-generalized Sidon sets in [n] is 2^{Θ(}√n) . Our approach is based on some variants of the graph container method.

Original language | English (US) |
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Pages (from-to) | 3-21 |

Number of pages | 19 |

Journal | Acta Scientiarum Mathematicarum |

Volume | 87 |

Issue number | 1 |

DOIs | |

State | Published - 2021 |

## Keywords

- Generalized sidon set
- The graph container method

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics