### Abstract

For r≧2 let p(n, r) denote the maximum cardinality of a subset A of N={1, 2,..., n} such that there are no B⊂A and an integer y with {Mathematical expression}b=y^{r}. It is shown that for any ε>0 and n>n(ε), (1+o(1))2^{1/(r+1)}n^{(r-1)/(r+1)}≦p(n, r)≦n^{e{open}+2/3} for all r≦5, and that for every fixed r≧6, p(n, r)=(1+o(1))·2^{1/(r+1)}n^{(r-1)/(r+1)} as n→∞. Let f(n, m) denote the maximum cardinality of a subset A of N such that there is no B⊂A the sum of whose elements is m. It is proved that for 3 n^{6/3+e{open}}≦m≦n^{2}/20 log^{2}n and n>n(ε), f(n, m)=[n/s]+s-2, where s is the smallest integer that does not divide m. A special case of this result establishes a conjecture of Erdo{combining double acute accent}s and Graham.

Original language | English (US) |
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Pages (from-to) | 297-306 |

Number of pages | 10 |

Journal | Combinatorica |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Keywords

- AMS subject classification (1980): 10A50, 10B35, 10J10

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## Cite this

*Combinatorica*,

*8*(4), 297-306. https://doi.org/10.1007/BF02189086