### Abstract

Suppose ε > 0 and k > 1. We show that if n > n_{0}(k, ε) and A ⊆ Z_{n} satisfies |A| > (( 1 k) + ε)n then there is a subset B ⊆ A such that 0 < |B| ≤ k and Σ_{b∈B}b = 0 (in Z_{n}). The case k = 3 solves a problem of Stalley and another problem of Erdös and Graham. For an integer m > 0, let snd(m) denote the smallest integer that does not divide m. We prove that for every ε > 0 there is a constant c = c(ε) > 1, such that for every n > 0 and every m, n^{1 + ε} ≤ m ≤ n^{2} log^{2}n every set A ⊆ {1, 2,..., n} of cardinality |A| > c · n snd(m) contains a subset B ⊆ A so that Σ_{b∈B}b = m. This is best possible, up to the constant c. In particular it implies that for every n there is an m such that every set A ⊆ {1,...,n} of cardinality |A| > cn log n contains a subset B ⊆ A so that Σ_{b∈B}b = m, thus settling a problem of Erdös and Graham.

Original language | English (US) |
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Pages (from-to) | 196-205 |

Number of pages | 10 |

Journal | Journal of Number Theory |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1987 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*27*(2), 196-205. https://doi.org/10.1016/0022-314X(87)90061-8