### Abstract

It is shown that every set ofnintegers contains a subset of sizeΩ(n^{1/6}) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for everyε>0 and everym>m(ε) the following holds. IfA_{1},...,A_{m}aremsubsets of cardinality at leastm^{1+ε}each, then there area_{1}∈A_{1},...,a_{m}∈A_{m}so that the sum of every nonempty subset of the set {a_{1},...,a_{m}} is nonzero. This is nearly tight. The proofs of both theorems are similar and combine simple probabilistic methods with combinatorial and number theoretic tools.

Original language | English |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 86 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 1 1999 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics